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What do you need to know about Oscillations and Waves?

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Thermal physics provided the transition from macroscopic physics to microscopic physics, and electricity is all about electrons, so why are we backtracking now to talk about bouncing springs and wiggling strings?

 

It’s because waves are going to be essential, ultimately, to our understanding of the inner workings of the atom. Bohr’s earliest thoughts about the atom involved viewing electrons as little mass/spring systems – “harmonic oscillators” – inside the atom. Later he would switch to a model in which the electrons orbit the nucleus.  Orbits are circular motion, and circular motion is intimately related to simple harmonic motion.  It all ties together!

 

Key Points for IB Students
  • F = -kx or a = -kx are the defining equations of simple harmonic motion.  Make sure you know what these graphs look like (straight line, negative slope, through origin)
  • Sound is a longitudinal wave
  • Electromagnetic waves (light) are the only type that need no medium.
  • Light must be a transverse wave, because we can polarize it
  • Know how to derive velocity = frequency * wavelength
  • Be sure you understand transverse and longitudinal waves
  • Be able to use and sketch graphs of displacement vs time AND displacement vs position.  From a graph of displacement vs position, you can get wavelength but not period.  From a graph vs time you can get period but not wavelength.  This is a common source of trouble for students.

Image result for displacement time graph wavelength

 

  • A graph of displacement vs time or position looks the same regardless of whether the wave is transverse or longitudinal.  It’s only the direction of the displacement that is different.  For transverse, displacement is perpendicular to the direction the wave moves.  For longitudinal, displacement is parallel to direction of travel of the wave.
  • Energy of a wave is proportional to square of amplitude (because energy in SHM is proportional to square of displacement, as we see in energy of a spring = 1/2kx^2)

 

Oscillations and Simple Harmonic Motion (4.1)

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Did you ever wonder why pi shows up in so many places where you might not expect it? One reason is that simple harmonic motion is related to circular motion.  Pi is not just about circles, it’s about oscillations, and oscillations are everywhere, so pi is everywhere.

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Image result for oscillation

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Image result for simple harmonic motion

Watch the following video!

 

Traveling Waves (4.2)

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Wave Characteristics (4.3)

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a) Wavefronts

This simulation can help you visualize wavefronts: http://phet.colorado.edu/en/simulation/wave-interference

b) Polarization

Simulator for polarized light passing through polarized film:  http://tutor-homework.com/Physics_Help/polarized_light.html

 

Watch this Youtube video showing use of polarized light to identify stress in molded plastic

 

c) Superposition

Good animation of superposition of waves  http://www.acs.psu.edu/drussell/Demos/superposition/superposition.html

 

Wave Behavior (4.4)

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a) Refraction

Simulator for refraction, critical angle, and total internal reflection: http://phet.colorado.edu/en/simulation/bending-light

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b) Diffraction

  • Single slit diffraction simulator   This sim allows you to see how slit width and wavelength affect the diffraction pattern:  http://www.walter-fendt.de/ph14e/singleslit.htm
  • Double slit diffraction demonstration by Veritasium.  Real life demonstration of Young’s famous experiment that “proved” light is a wave

 

  • Double source interference simulator.  This is not an example of diffraction but it does give you a very good picture of what the interference pattern looks like when you have two sources of waves (Since slits are like independent sources, this is the same pattern you get when studying double slit diffraction) http://phet.colorado.edu/en/simulation/sound

 

Standing Waves (4.5)

Wave on string demo showing multiple harmonics with a vibrating string

 

a) Understanding standing waves as superposed reflections: http://www.acs.psu.edu/drussell/Demos/superposition/superposition.html

b) Longitudinal waves in pipes

Image result for longitudinal waves in pipes

 

 

 

Measuring speed of sound  using standing waves

 

 

c) Resonance

 

Famous video of Tacoma Narrows bridge collapse video

 

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Millenium bridge resonates shortly after opening

What do you need to know about Thermal Physics?

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I love physics, but I used to think this one topic, thermal physics, was boring.  I had to teach it twice before I understood it well enough to see how interesting it really is!

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What makes thermal physics interesting is that it provides a link between the macroscopic world of Newton’s laws and the invisible microscopic world of atoms and molecules.

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Thermal physics provided the vital clues that led to our discovery of the atom. It also provided the first hint of the quantum revolution to come, by introducing the concept of unpredictability and randomness into the field of physics.

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Brownian motion is random and unpredictable, even though we may imagine that it results from an uncountable number of simple Newtonian collisions.

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Image result for brownian motion

 

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It would be up to Einstein to show that Brownian motion provides irrefutable evidence of the existence of atoms.

He did that in his “miracle year”, 1905, when he also published his theory of special relativity.

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Einstein came up with the world’s most famous equation,

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E = mc^2

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He proposed an explanation for the photoelectric effect, thereby launching the new field of quantum mechanics.

It’s remarkable that Einstein only won one Nobel, and he didn’t even win that one until 1921.

 

Video lessons for you!

You will need to Download All Files (upper right) and then click on the one called container.html

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Calorimeter simulation to measure specific heat capacity

http://www.chm.davidson.edu/vce/calorimetry/SpecificHeatCapacityofCopper.html

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“The Last Question” 

Isaac Asimov short story about entropy, http://www.multivax.com/last_question.html

I predict that if you read this short story, you will remember it decades from now.  It packs a lot of punch.

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Watch the videos below on the misconception about heat and temperature.

 Misconceptions about heat, Veritasium video

 

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Misconceptions about temperature, Veritasium video

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The Dippy Bird!

What do you need to know about Electric Charge and Field?

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There are only four fundamental forces in physics, and in high school physics we will learn about three of them.  These forces are called “fundamental” because we don’t have a deeper explanation for why they exist.  The four are

  1. Gravitational Force
  2. Electric Force
  3. Strong nuclear force
  4. Weak nuclear force (we won’t learn about this one)

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Gravitational force

Gravity is the most familiar of the four.  Even though gravity is relatively weak, it has a big impact on us because it is an attraction between masses, and we happen to spend our entire lives in very close proximity to an enormous, enormous mass:  planet Earth.

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Image result for gravitational field

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Gravity is so familiar to us that humans observed its effects – things fall down –  for thousands of years before anyone really recognized it for what it is.

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Isaac Newton was the one to realize that the force that makes apples fall from trees is the same force that makes the moon travel in an elliptical (nearly circular) path around the earth, and the earth follow an elliptical (nearly circular) path around the sun.

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Newton deduced that gravitational attraction between point masses must follow an inverse square law because that’s how you get elliptical orbits, and he knew that Kepler had already determined that the orbits of the planets were all elliptical.

In studying gravitational fields, we learn that gravitational fields are responsible for the force that we call weight, and we can calculate weight using F = mg.

We learned Newton’s law,

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F = GMm/r^2

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It tells us the force of attraction between two point masses, or between two spherical masses, if we measure from the center of one sphere to the center of the other.

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Image result for F = GMm/r^2

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We learned that we can use the concept of an invisible field to understand action at a distance, and we learned to characterize the strength of that field by considering the force per unit of mass that it exerts on a test mass placed in the field.

We learned that because massive objects have weight, they gain potential energy when we displace them in opposition to a gravitational field.  That’s a fancy way to say that the object gains energy when we lift it up.  delta PE = m*g*delta h, where h is the height of the object.

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 What does all this have to do with electric charge and electric fields?

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Electric field

The easiest way to build your understanding of electric fields is to recognize how similar they are to gravitational fields. Once you know the concepts of one, you know the concepts of the other.  When you can analyze one, you can analyze both.

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Gravity is an attraction between masses.  What is mass?  It is a fundamental property of a particle or object.  It tells us how hard gravity will pull on that object.

The electric force is just like gravity, but instead of operating on mass, it operates on another fundamental property that a particle or object has: charge.  Charge tells us how hard the electric field will pull on an object.

Image result for electric field

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Like gravity, the electric force follows an inverse square law that has exactly the same form as Newton’s law.  We just replace mass with charge.

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We define electric field strength as the force per unit of charge it exerts on a positive test charge placed in the field.a

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That points to an important difference between gravitational force and electric force:

  • There’s only one kind of mass, but there are two kinds of charge.

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  • We’ve arbitrarily named them positive and negative, but we could have just as well called them red and blue.

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  • Unlike charges attract, the way masses attract each other with gravity, but like charges repel each other, and we don’t see that kind of effect with gravity.

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Here now is a copy of the paragraph above about the gravitational force, adjusted so that it is about the electric force:

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In studying ELECTRIC FIELDS, we learn that ELECTRIC FIELDS are responsible for the force BETWEEN TWO CHARGES, and we can calculate THAT FORCE using:

F = qE  

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Image result for ELECTRIC FIELDS

We learned COULOMB’s law,  F = kQq/r^2.  It tells us the force of attraction between two point CHARGES, or between two spherical CHARGES, if we measure from the center of one sphere to the center of the other.

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We learned that we can use the concept of an invisible field to understand action at a distance, and we learned to characterize the strength of that field by considering the force per unit of CHARGE that it exerts on a test CHARGE placed in the field.

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We learned that because CHARGED objects EXPERIENCE ELECTRIC FORCE, they gain potential energy when we displace them in opposition to an ELECTRIC field. That’s a fancy way to say that the object gains energy when we MOVE IT THROUGH A POTENTIAL DIFFERENCE.  delta PE = m*delta V, where V is the ELECTRIC POTENTIAL AT THE LOCATION of the object.

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See the similarity? Gravity is more intuitive because you experience it all the time, but the electric force works mostly the same way on charges as gravity works on masses.

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There are some differences too. There are two types of charges but only one type of mass, so the electric force can repel or attract, while gravity can only attract.

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The electric force is much, much stronger than the gravitational force.  How can we compare them?  Consider the magnitude of the electric force between two electrons, and compare it to the gravitational force between them.

The electric force is more than 40 orders of magnitude larger!  Ironically, the strength of the electric force is the reason we rarely notice it.  Because it is so strong, positive and negative charges are almost always paired together, so most objects are neutral, making it hard for us to notice the effect.

But consider what happens when you pull a comb through your hair, or rub a balloon with some fur.  You can use the comb or balloon to pick up scraps of paper. That might not seem so impressive until you realize that the little charge on the comb or balloon is exerting a force on the paper that is stronger than the gravitational force exerted by on the paper by the entire planet!

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The electric force is so strong that we don’t notice that is is actually the force that is at work when we push or pull on something.  What we think of as touch is really a manifestation of the electric force!  When we push on a chair, if we look at a microscopic level, what’s really going on is that the electrons in our hand are repelling the electrons in the book.  Our atoms are neutral but the electrons are on the outside and the protons are on the inside, so if you bring atoms close together the way they bump into each other is via the repulsion of their electrons.

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I like to make a chart to show the similarities (and differences) between gravity and the electric force

This illustration helps complete the analogy between the two types of fields:

 

 

 

I love this clip of Richard Feynman talking about the electric force

 

 

Here’s a nice video that helps visualize the shape of the electric field

Interview with The Smart Local

4 Ways Tuition Is Done Differently at The Physics Cafe

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For secondary school and JC students, mugging the examination syllabus can sometimes feel like an uphill battle; it may be tough, but it’s definitely not impossible. Subjects like Physics and Mathematics tend to get super convoluted and classroom lessons can only do so much to help students make it less so.

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That’s where a premium tuition centre like The Physics Cafe comes in. Not your run-of-the-mill centre, it’s got a yearly enrolment of 1,000 students, plenty of student services and a unique learning approach that’s helped at least 8 out 10 of their students achieve their ‘A’ grades. Here’s how they do it:

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Here’s how they do it:

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1. Lessons are dynamic and you can watch them online like a uni student

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We’ve all experienced the occasional “which page is the teacher on?” moment during school time. At The Physics Cafe, you’ll find that this never happens as the lessons are engaging throughout – student participation plays a big part here.

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Better yet, all of the lessons are recorded and made available online via a digital library for students’ personal revision. Students can easily book the digital lesson that they want to watch through the website, in preparation before major exams.

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The catalogue in their digital library is extensive – you can access all of the lessons that you’ve been through, for revision.

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The best part about this “digital cafe” is that the lessons are open to non-students too. You’ll have to come down to the tuition centre’s study area to stream the lessons, but for only half the price of a live lesson and complimentary access to the cafe’s pantry, it’s well worth the trip.

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All you have to do is book a timeslot here.

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2. To be a tutor here, you need to go through a Ninja Warrior-like process

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The Physics Cafe tutors are endearingly termed “chefs” but they may as well be regarded as Masterchefs of Physics. For every 30 to 40 applicants interviewed, only one is hired. And the process doesn’t just stop there – even if they’re experienced, they’ll have to go through a year’s worth of training conducting mock lessons before officially teaching a class.

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The centre’s founder Mr. Dave Sim can attest to this rigorous process – having taught at Raffles Junior College (now Raffles Institution) for six years, he understands the importance of being committed to his students and making Physics as accessible as possible. A quick look through their whopping list of testimonials – categorised by school – will show the tutors’ impact on their students.

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Mr. Dave Sim and his fellow tutors maintain a close relationship with their students, beyond the classroom.

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Don’t expect a spammage of notes or photocopies of Popular-bought assessment books here either – like an actual school syllabus, the tutors spend time putting together booklets of notes that are easier to understand than school textbooks. They’re highly sought after by non-students too!

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Top-secret notes!

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 3. Lessons are not a repetition of what you’ve learnt in school

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If you’re hesitant about tuition classes, it’s probably because you’re worried about spending an extra 2 hours of school, after school. But at The Physics Cafe, you won’t learn the same thing twice.

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“Simple harmonic motion” may not sound so simple during an 8am lecture, but when you sit in here, you’ll get it so much quicker. Lessons – often 2 hours – consist of two concise parts: understanding concepts, where concepts are broken down and explained, and problem-solving, where you get to apply the concept on real problems.

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Bonus: If that still doesn’t help, sometime the tutors will carry out live demonstrations!

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Their June holiday crash courses are popular among students for being major confidence boosters. These super comprehensive 4-hour lessons – with pizza breaks in between! – recap only the things that you need to know before a major exam. No longer will you have to rely on luck by spotting questions and learning more than you actually have to.

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Pizza during breaks? Yes, please! 

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4. The tuition centre is a legit cafe that provides free food AND free travel

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The pantry is constantly stocked. There’s even a free-flow vending machine!

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The name “The Physics Cafe” isn’t a marketing gimmick. It’s really akin to a hipster cafe – with a foosball table and a free-for-all pantry that rivals even that of TSL’s. If it wasn’t for the fact that it’s a tuition centre, we could have featured it in this month’s listicle of new cafes to try.

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Students can indicate on the map the destination they wish to go to

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And  the perks don’t just end at F&B – the cafe provides free shuttle rides to Botanic Gardens, Jurong East, Serangoon and Paya Lebar MRT. And if you’re one of those staying back late for revision, they’ll even book a Grab or Uber ride to send you home – free of charge!

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The Physics Cafe – A different kind of tuition centre

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Most people brag about results, but only the students here get to brag about results and how cool their tuition centre is. With ergonomic chairs for every student, high-tech equipment and a space that’s half the size of a football field, tuition here definitely feels like less of a bore.

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And the results speak for themselves too with the​ centre’s impressive track record​:

  • In 2016, at least 80% of their students scored A in Physics and/or Maths in the ‘A’ level examinations, double that of the national average
  • 9 out of 10 of their students scored a distinction in the Physics and/or Maths in the O-level exam
  • Check out this list of students who scored distinctions. You may even recognise some of your seniors!

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The Physics Cafe is already one of the most established tuition centres around – with over 1000 yearly student enrolment – and in the coming year, more students will be able to benefit from their lessons. In addition to their Toa Payoh and Beauty World branches, two new centres are expected to be opened in Novena and Marine Parade in early 2018.

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Spots are high in demand so registration is online-only.

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Bonus: Look out for their ambassador programme, officially launching in late 2017. It’s an initiative that provides students with partial and full scholarship to attend their tuition classes!

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Sign up with The Physics Cafe now​!

 

This post was brought to you by The Physics Cafe.

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Read more at TheSmartLocal website: http://thesmartlocal.com/read/the-physics-cafe

 

What do you need to know about Forces & Newton’s Laws of Motion?

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A force is just a push or a pull

There are lots of ways to push or pull:

  • Gravity pulls objects down toward the ground.  We call that force the weight of the object.

 

  • A string or rope can pull on an object.

 

  • Springs can do that, too.  That’s tension.

 

  • One object can push on another.  That’s a contact force.  We’ll be particularly interested in the normal force (or the reaction force).

 

  • An object might float in liquid.  The object is supported by a force called buoyancy, or upthrust.

 

  • Electric charges attract or repel each other.  That’s the electric force.

 

  • Air resistance opposes motion through the air.  That’s called a drag force.

 

  • Friction opposes the motion of one object sliding against another.  We have two main kinds of friction force:  static and kinetic (or dynamic)

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Even though there are a lot of ways to push or pull, there are only four FUNDAMENTAL forces in physics.  “Fundamental” means we don’t have a deeper explanation for them. The four are:

  1. Gravitational force
  2. Electromagnetic force
  3. Strong nuclear force
  4. Weak nuclear force

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Image result for gravitational force pictures

Gravitational force

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Image result for electromagnetic force

     Electromagnetic force

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You have only ever had direct experience with the first two.  EVERY force you have EVER experienced or witnessed is one of these two.

You’ve never directly seen the nuclear forces because they operate only inside the nucleus.

But wait: where’s tension, and friction, and contact force on this list?

Those are all instances of the electromagnetic force!

When a solid object pushes against another object, at the atomic level what’s really going on is that the negative electrons of the atoms of one object are repelling the negative electrons of the atoms of the other object.  The reason a string can exert tension is because its molecules are held together by electric forces.  Drag and friction also arise because of the interactions of atoms bumping into each other, by which we mean experiencing electron vs. electron repulsion.

In high school physics, we’ll learn a lot about gravity and the electromagnetic force, and a little about the other two.

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Newton’s Laws of Motion: How to remember which law is which

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1) The pen on my desk is not going to move unless I push on it.  That’s inertia.

2) If I do push on it, the acceleration will depend on how hard I push and how much mass the pen has.  F = ma.

3) However hard I push on the pen, the pen will push back on me.  Equal and opposite reactions.

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Important ideas

There is no difference between uniform motion and being at rest.  Motion is relative.  That’s why “an object at rest will stay at rest” is THE SAME AS “an object in uniform motion will stay in uniform motion“.  Rest is uniform motion.

Third law force pairs are on different objects, not the same objects.  That’s why they don’t “cancel out”.

Third law force pairs have to be the same type.  Gravity and gravity, contact force and contact force, electric force and electric force.

An object at rest on a table has a normal force equal to its weight.  That’s an example of the second law, F=ma.  The forces add up to zero so the acceleration is zero.  It is not an example of the third law.

It’s true in this case that the normal force is equal to the weight, but it does not have to be. If the table breaks, the normal force will be less than the weight and the object will accelerate downward, but still the third law will hold:  the object will be pushing on the table just as hard as the table pushes on the object.  Both of those forces will be less than the object’s weight!  What is the third law pair for the weight of the object?  It’s the pull of the object on the earth.

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Understanding Newton’s First Law (Law of Inertia)

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Image result for newton first laws of motion

European Space Agency video aboard International Space Station:

 

Veritasium: Why Does the Earth Spin?

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Veritasium:  How Does the Earth Spin?

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Newton’s Second Law (F = ma)

ESA video aboard ISS

 

Newton’s Third Law (equal and opposite reaction force pairs)

 

What do you need to know about Free-fall, Projectile Motion, Air Resistance, and Terminal Velocity?

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Free-fall:  a special case of uniformly accelerated motion

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When you drop a hammer, it falls to the ground.  While it is falling, the only force acting on it is gravity. (There’s also a little bit of air resistance, but for a hammer moving at relatively low speeds, we can ignore that).  We call this state of motion “free fall“.

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Galileo was the first to figure out that objects in free fall all accelerate at the same rate.

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He didn’t have the technology to measure this precisely, but he could drop objects of different weights from a great height and see that they hit the ground at the same time.  He also performed a “thought experiment” that convinced him that heavy objects must accelerate at the same rate as lighter ones (still ignoring air resistance).

He imagined a big rock and a little rock tied together with a short rope.

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  • If heavy objects fall faster than light ones, then if we drop the pair of rocks we should see the big rock trying to fall faster, pulling on the rope, with the smaller rock lagging behind, slowing it down.  The two rocks should accelerate at a rate in between the rate that either one would accelerate on its own.

 

  • But when we tie the rocks together, we turn them into a single object whose weight is greater than either rock on its own.  The two rocks should accelerate at a rate that is faster than either one would accelerate on its own!

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So which is it?  Do the tied rocks fall slower or faster than the big rock would fall by itself?

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Galileo recognized that the only way this conflict does not exist is if the light rock and the heavy rock both accelerate at the same rate, regardless of whether they are tied together or not.  Even a feather will fall at the same rate as a hammer, if there’s no air resistance.  Don’t believe it?  Watch!

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Feather and Hammer Drop on the Moon

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Bowling Ball and Feather in world’s largest vacuum chamber

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Heavy objects fall at the same rate as light ones.

 

We’ll learn WHY this is true in a later note. For now, we can say that physicists have determined with a fair degree of precision that for an object in free fall near the surface of the earth, the acceleration due to gravity, which we call g, is 9.8 (m/s) / s, or 9.8 m/s^2.

If you drop a rock, after 1 second it will be going 9.8 m/s. After 2 seconds it will be going 19.6 m/s. And so on.

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The value of g actually varies a little bit from place to place, with a maximum of 9.83 and a minimum of 9.76 m/s^2.  If you are doing AP Physics, you can round to 10.  The physics teachers I know use 9.8 or 9.81 m/s^2.

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Projectile Motion

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An object in free fall is called a projectile.  Some of the earliest important applications of physics involved projectiles and were studied carefully at military academies, where students learned to predict the paths of stones launched by catapults or trebuchets or cannons.

Image result for projectile motion with air resistance

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Naturally no one wanted to just shoot cannonballs straight up in the air, so they needed to understand motion in 2 dimensions, not just one.  The cannonball has to go up in the air but it also has to travel horizontally.

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In the absence of air resistance, the horizontal motion is completely independent of the vertical motion.

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If you understand vector addition well you will see why this must be true. Take vector V and vector H, at right angles to one another.  Their resultant is C.  Make V larger or smaller.  Does the H-component of the resultant vector change at all?  Nope. Make H larger or smaller.  Does the V-component of the resultant change?  Nope.  They’re independent.

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Image result for vector diagram

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This is pretty convenient!  It means that if we know the initial velocity vector (speed and direction), we can know everything about the path of the projectile.

We find the vertical velocity component and apply our equations for accelerated motion, using g for acceleration.  That will tell us how high the projectile goes, and how long it is in the air.

Knowing how long it is in the air, we can use the horizontal component of the velocity to find out how far it goes.  There’s no acceleration in the horizontal direction because there’s no force acting to change the horizontal speed.

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Imagine you have a cannon aimed so that you can shoot a cannonball exactly horizontally. You also rig up your cannon in such a way that another cannonball drops straight to the ground at the very instant you fire the horizontal cannonball.  Which cannonball will hit the ground first?

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They land at the same time.  Both balls are affected by gravity from the moment they are released by the cannon.  Gravity does not ignore the horizontal cannonball just because it is going really fast.  It pulls it down at exactly the same rate as the ball that was simply dropped.

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This is often called the monkey-and-hunter problem, because instead of cannons the story often involves a hunter firing horizontally at a monkey hanging from a tree.  The monkey let go at the same instant the gun is fired, but sadly the monkey’s fall puts him right in the path of the falling bullet.  You might find this pretty easy to understand once you have thought about the independence of horizontal and vertical movement.

But what if we make it more complicated?  Instead of aiming horizontally, the hunter has to aim upward because the monkey is high up in the tree.  The monkey still lets go the instant the gun is fired.  Does he still get hit?  Let’s see:

 

Monkey and Hunter Demonstration

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Air Resistance and Terminal Velocity

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Real projectiles launched on earth experience air resistance.

At low speeds, for objects with a high ratio of weight to surface area (like, not a feather or a sheet of paper), we can ignore air resistance.

At high speeds, even an aerodynamic bullet experiences a lot of air resistance.

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Air resistance is a fairly complicated force.  We aren’t going to try to analyze it in detail.  But we can make a couple of important observations about it.

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  • First, we can say that adding air resistance is going to lessen a projectile’s maximum height.

That means it will be in the air for a shorter time, so it can’t go as far. And the air resistance is also going to cause the horizontal component of the velocity to decrease as time goes by. That shortens the range of the projectile even more.  The path of the projectile is no longer a parabola.  The descent is much steeper than the ascent.

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  • Second, the force of air resistance varies with speed, a lot.

When you double the speed of a ball, the air resistance doesn’t just double, it goes up by a factor of perhaps 4 (there’s no simple way to predict the exact number).  This explains the existence of a phenomenon we call terminal velocity.

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If you jump out of an airplane, at first you fall with an acceleration of 9.8 m/s^2.  But as your speed increases, the air resistance increases, so your acceleration quickly becomes less than g.

Still, you are speeding up, but that means that still, the air resistance is increasing.  So your acceleration continues to decline.  Where does this stop? Before long you will reach a speed where the upward force of air resistance matches the downward pull of gravity.  You’ll be falling very fast at this point, but you won’t be accelerating anymore. You’ve reached terminal velocity.  For a human in the air, terminal velocity is about 120 miles per hour.  So you can’t rely on it to save you if your parachute doesn’t open.

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Terminal velocity depends a lot on the shape and weight of the object.

a

When you open your parachute, you effectively change your shape, and your terminal velocity becomes much smaller, so you slow down until you reach the terminal velocity permitted by the chute.  This should be a speed that is low enough to allow you to land safely.

You can watch the effect of terminal velocity in this breathtaking video of Felix Baumgartner jumping from space.

He’s so high up at the beginning that he is in free fall, with no air resistance.  His speed increases very quickly.  As he falls, he encounters increasingly dense air.  His speed is so great that even this thin air eventually stops his acceleration.  Watch the numbers on the screen and listen to the mission control guy to see how fast he is going at that point.  It’s a lot faster than 120 mph!

 

Felix Baumgartner Jumps from Space (Fast forward to about 57:00 into the video for the good stuff)

 

 

Alan Eustace Outdoes Baumgartner!

 

What do you need to know about Measurement & Uncertainty.

Yes we know. Fascinating and intriguing may not be the first descriptors that come to mind at the very mention of this topic. In fact, and also very unfortunately, it has been a sufferer of chronic neglect by generations and generations of students (and even teachers) 🙁

Measurement and Uncertainty Physics FAQ

Mundane as it may seem, we cannot discredit this topic for laying out the foundations to classical and modern Physics. On a more personal note, it is thanks to this topic that students and teachers alike can make sense of scientific experiments, mind-boggling exam questions etc… As such, we would like for you to embrace this poor ignored topic,  and perhaps reverse your preconceived notions about it. Let us show you that Measurement and Uncertainty can be a joy to learn about!

The larger scale of things…

Now first, let us put things into perspective for you… Actually, we’ll just let this applet do the talking.

The Kilogram

Also, DID YOU KNOW that the kilogram is defined by a physical artifact, and in fact, is the only SI unit that is so? Here’s a video on the scientific community’s efforts to eradicate the kilogram’s dependence on a physical object.

But wait a minute, it is also highly probable that this omnipresent unit may be rendered obsolete in future! Scary isn’t it, to fathom that this has the potential to upset the achievements of previous well-established scientific work, and corrupt our current understanding of the world.

Apart from SI units, measurering apparatus and equipment are big stars in this topic, so let’s zoom in onto them. The infamous vernier caliper, micrometre screw gauge, and other equipment make their debut typically as this point. To make things easier for you, we even found a vernier caliper simulator so that you can practise using one! Trust us, we know how easy it is to forget how to use one of these, so here it is to keep it fresh in your minds: Vernier caliper simulator

In essence, this topic is all about data – collecting, recording, and last but not least, analysing them. Plotting graphs is an important skill that physics students need to grasp in order to make sense of data. Here’s an interesting read on the significance of graphs and trends: Why we graph data instead of trusting other statistical measurements

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That’s all we have for you for now. Stay tuned to find out what more we have to offer! We hope you’ve had a good read:)

 

 

What do you need to know about Kinematics?

Defining Velocity

Velocity is the rate of change of position.

If you go in a straight line from A to B, your velocity is just the distance between A and B divided by the time it took to go from A to B.

e.g. If the straight-line distance is 100 m, and it takes 20 seconds, your velocity is 5 m/s.

The units of velocity have to be length / time : m/s and km/hr are pretty common.

Mathematically, we can say

v = Δs/Δt.

somtimes

Sometimes teachers will just say v= s/t.

What does the Δ mean? It is the Greek letter delta, and it means “change in”.  (delta is like our letter d, and d here stands for difference).

So velocity is (change in displacement) / (change in time). If initial displacement and initial time are both zero, then Δs/Δ t is the same as s/t.

So if you start from 0, then Δ s is the same as s, but if you start from somewhere else, then to be very precise we want to recognize that velocity is related to the CHANGE in the displacement.

  • If you move from 0 to 5, your new displacement from 0 is 5. So your change in displacement is 5.
  • If you move from 5 to 8, your new displacement FROM 0 is 8, but the change in your displacement is 3.

A more

A more complicated way of writing the definition we already wrote is:

(Final displacement – initial displacement )/ time

or (Sf – Si)/t

or worse yet (Sf – Si) / (Tf -Ti).

Yuck!

Don’t let this confuse you if you encounter it.

You need to memorize this definition:

 

Velocity is the slope (or gradient) of the displacement vs time graph

 

 

Image result for displacement vs time graph represents

 

  • If velocity is constant, then the graph will just be a straight line, because straight lines have the same slope everywhere.
  • If the s vs t graph is not a straight line,we can still find the velocity at any particular instant in time by drawing a tangent line to the curve at that point and measuring its slope.  Or we can find the average velocity for any period by just drawing a straight line between any two points on the curve and measure that slope.

 

Defining acceleration

Acceleration is rate of change of velocity. 

We can write a = Δ v / Δ t. Units of (velocity / time) is (m/s)/s or  m/s^2.

 

If you are in your car going straight at 20 km/hr, and you speed up to 30 km/hr, your velocity changed by 10 km/hr. If you made that change in 5 seconds, then your acceleration was (10 km/hr) / ( 5 seconds) = (2 km/hr)/s.

If the change in velocity is negative, then you have negative acceleration. Negative acceleration could mean you are slowing down, and we still call that acceleration. But it could also mean that you are speeding up in the negative direction!

Let’s take an example of a ball thrown in the air. We decide to call the direction “up” positive.

 

Image result for acceleration of ball thrown up

 

 

In the beginning the ball has positive velocity. Is it accelerating on the way up? YES IT IS!

It is slowing down, so its velocity is changing. By definition, changing velocity is acceleration. The acceleration here is negative, because it is downward, and we have already defined up as positive. That negative acceleration is causing the velocity to become less and less positive as time passes. The ball is slowing down.

a

   Image result for acceleration of ball thrown up and going down

 

a

At the top of the arc, the velocity is zero. The acceleration continues, though, so the velocity passes right through zero and becomes negative, and then it gets bigger and bigger in the negative direction.

The ball is speeding up! The acceleration – the rate of change of the velocity – was the same the whole time, and it was ALWAYS downward. There was no time after you let go of the ball when the direction of the acceleration was anything other than downward. The result of that downward acceleration was a ball that was at first slowing down and then speeding up! But a better way to think about it was that the ball’s velocity was changing in the negative direction the whole time.

 

The definition of acceleration is another definition you just have to learn. a = Δv/Δt. If you want to learn physics, you’ve got to just know this.

Acceleration is the gradient (or slope) of the velocity vs time graph

if velocity

  • If velocity is increasing at a steady rate, then acceleration will be a positive constant.
  • If velocity is decreasing at at steady rate, then acceleration will be a negative constant.
  • If velocity is changing, then acceleration is constant at 0.

 

Obviously you can go the other way, too:

  • If acceleration is a positive constant, then the velocity graph is a straight line tilting up;
  • if acceleration is a negative constant, then the acceleration graph is a straight line tilting down,
  • and if acceleration is 0, the velocity graph must be flat.

Be sure not to confuse zero acceleration with zero velocity.

Zero acceleration means velocity is not changing.  It could be 0 and not changing, but it could also have any non-zero value without changing.

Image result for velocity vs time graph different acceleration

 

Image result for velocity vs time graph different acceleration

 

The kinematic equations

The useful set of equations that we can use for uniformly accelerated motion are sometimes called the SUVAT equations.

s = displacement (some teachers use x or Δx)

u = initial velocity (some teachers use Vi or Vo)

v = final velocity (some teachers use Vf)

a = acceleration

t = time

 

 

s = (u + v)*t/2

This one is easy.  It’s just a recipe for calculating average velocity and multiplying it by time to get displacement.

It only works if acceleration is constant, though.

You should already know that the average of 7 and 9 is (7+9)/2, so this equation should not be hard to understand, derive, or memorize.

 

v = u + at

Ending velocity is just initial velocity plus the change that comes from accelerating for some period of time.

This one should also be pretty easy for you to derive or memorize.  It comes right from the definition of acceleration.

 

s = ut + 1/2*a*t^2

This one is not so obvious, but it tells you how to find displacement of an object that is already in motion and then accelerates.

The object gets some displacement just from the fact that it was already moving (that’s the ut part), and then it gets turbo-charged by the acceleration (1/2*a*t^2).  This one is a little harder to memorize, but worth it.

 

v^2 = u^2 + 2as

Not obvious or intuitive at all, right?  I rarely use this one.

Anything you can do with this one you can also do by combining two others.  I didn’t get around to memorizing this one for years.  (It is useful, though, in deriving the equation for kinetic energy:  KE = 1/2*m*v^2).

 

*Note that velocity, acceleration, and displacement are all vectors, but we are only dealing here with motion in one dimension, which means objects can only move forward or backward along a straight line.

We will have only two directions:  up and down, or left and right, or east and west, or positive and negative, or some other pairing of opposites.

It is very important to remember that these equations only apply for uniform (constant) acceleration. Constant acceleration can include zero acceleration, of course.

 

Where do those SUVAT equations come from?

We can derive all of the equations just from the definition of velocity and acceleration!  To see the derivations, go to this link and page down about half way:

https://www.khanacademy.org/science/physics/one-dimensional-motion/kinematic-formulas/a/what-are-the-kinematic-formulas

 

A lot of times, knowing how to derive an equation is not that useful when you are solving problems, but I encourage you to closely study the derivation of Δs = ut + 1/2*a*t^2, where they use the area under the curve, for two reasons.

  1. You can use the same technique to solve real problems.
  2. Second, you are going to see this same area-under-the-curve approach used again and again.

For example

For example when we figure out the energy stored in a spring ( 1/2*k*x^2) and when we figure out the equation for kinetic energy (1/2*m*v^2).

Do you notice the similarity in form between those 3?  It’s because they’re all derived by finding the area of a triangle.  A triangle’s area is 1/2*b*h.  If h is linearly related to b, you will always get something that involves 1/2*(some constant)*b^2.

 

If you don’t see why displacement is the area under the velocity vs time curve, this video will help, and you should watch it, because this is a really important and useful concept that we apply over and over.

The Maths Cafe – JC Maths Tuition

Congratulations!!!

Our graduated batch of 2017 has done very well, bringing back stellar results, with:

Close to 10 out of 10 scored A or B in H2 Maths.

Close to 8 out of 10 scored A in H2 Maths.

 

Click here for the full listing of students with the results.

 

  

H2 & JC Maths Tuition

Your A levels are that final step before you move on to university education. As the tests get more frequent and the questions get tougher, you may find yourself falling behind on your grades. This is where The Maths Café comes in. As a team who have years of experience getting students to achieve high marks, our tuition provides you with the knowledge and skills that you can apply during those stressful final exams.

 

a level math tution

Ace your A Levels

If you’re afraid of falling behind, or simply want to ensure you get the best marks so you can assure a pathway into the university of your choice, register with The Maths Café today. Our courses will provide you with the skills to excel in all areas of math, including trigonometry, calculus, algebra and more.

And we don’t want to just sit you down at a desk and order you to work out complex questions on your own. At The Math’s Café, we work by providing you with questions and equations, breaking them down so you understand and getting you to work on them at your own pace. Once you’ve started at the bottom, we’ll gradually build your skills up until you’re confident in answering those really tough questions.

 

Get in touch with the experts at The Maths Café

Whether you’re looking for math tuition to improve your skills in a particular category, or need help building up your knowledge in all areas, we have a uniquely designed course available that you’ll love. Our teachers and councillors work as a team to provide students with a comprehensive understanding of what is needed to truly ace those exams.

So, don’t wait any longer, register today for classes at The Maths Café. If you have questions about maths tuition, fees or our teachers, don’t hesitate to contact our team, we’re happy to help you make the right choice.

We now find ourselves at the halfway mark of the year. It is a good time to consolidate your learning and prepare for the end of year examinations, or in the case of the J2 students, THE A level Examinations. Whether you live up to the legacies of the previous batches, or create one of your own is entirely up to how much effort and time you are willing to invest. And of course, we are here to help you with that. Under the guidance of our tutors and notes, you’ll find yourselves having an easier time! Scroll down to see what our students have to say about our classes.

 

 

Praises Corner

The Physics Cafe has helped me a lot for both Maths and Physics. Though I joined only in J2, in a short period of time, the classes here helped me to consolidate my learning and improve my understanding of the topics. The lessons are very engaging and easy to understand and the tutors help me with any questions I may have. Overall, I would really recommend the Physics cafe to anyone who needs help in gaining confidence in their own studies and help in understanding their subjects.
V.Meghana, RI

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The Maths Cafe has definitely helped me a lot in gearing me up to tackle challenging questions that aren’t typical/standard. This allowed me to gain more confidence in tackling any type of questions at hand and to hopefully attain >80 for the paper. The Secrets Enclosed Notes are also very useful as a form of summary for the whole syllabus and it really helped me in maintaining my A for Maths ever since I joined TPC and thereafter scoring even better than before I joined. Prelims had been tough and getting not only an A, but also somewhat my best Math results in J2 was a blessing and a booster for A levels. By scoring better in my H2 Maths, it also helped me in my H3 Maths (MOE) which I am currently undertaking.
Koh Wen Hao, JJC

 

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The Maths Cafe – O level/IP Maths Tuition

For students taking their O levels and major examination this year, you want to ensure that you can get all the marks in the bag. Maths and Physics are arguably the subjects that students can aim to score exceptionally well in. And if that’s what you’re aiming to do, you can count on us to take you there!

Our classes have produced batches after batches of students who performed OUTSTANDINGLY in their examinations – testimony of the effectiveness of our notes and tutors!

O level/ IP maths tuition

Secondary School A Maths Tuition

Mathematics has a wide range of uses in your day-to-day life. Not only is it a practical subject that can help you with future career ambitions in accounting, logistics and even engineering, but it also helps you with developing your ability to analyse and break down problems with complex solutions.

At The Math’s Café, our O level IP Maths tuition is designed to help students fine-tune their skills in calculus, fractions, algebra, trigonometry and more. Whether you’re struggling with one field or need help with it all, we’ll be able build up your skills and knowledge with teachings that you’ll actually enjoy.

 

 

 

Quality tutoring for IP Mathematics

At The Math’s Café, we don’t believe in sitting our students down and telling them to solve an equation. Instead, our approach is to understand the specific needs of students and help them build their skills from the bottom-up. Ensuring that students are masters of the bottom level will help them gradually and surely grow step-by-step

Our knowledgeable teachers are well equipped to train you to respond to questions and equations with confidence. After understanding your skills and the way you learn, we help you develop your skills so you can answer any possible questions that you’re faced with during exams.

A proven track record at The Maths Café

If you want to get those grades up and build your skills before you get to your A Levels, The Math’s Café is your go-to. Our teachers and student councillors are high-achievers and well-versed in methods that will help you ace your exams.

We endeavour to help you get those well-deserved A’s, so register with us today and start developing your maths understanding with a friendly, dedicated team. If you have any questions about our lessons or tutors, don’t hesitate to get in touch before you register.

 

 

Praises from our students 😀

Better Understanding

Even though I joined The Maths Cafe halfway through the year, I have learnt many tips and formulas to apply during my maths paper. Now, I enjoy practicing math because I understand what the question is asking and how to apply whatever formulas I have learnt. Secrets Enclosed is a very useful guide to revision and has allowed me to recap on the different topics and brush up on my weaker ones. My peers in The Maths Cafe definitely have a better understanding of concepts and certainly have seem to be doing well in maths. The Maths Cafe has definitely allowed me to do well in both A.Maths and E.Maths and I would definitely recommend my friends to join this class 🙂
Matilda Woo, SST

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Making Revision Easier

The answering techniques taught to us by Ms Toh is very helpful as not only does it help us in during the thought process, but it also makes it easier for us to revise when reviewing our previous work. Having lessons in The Maths Cafe means having more attention from the teacher, which means that it is a lot easier for you to approach the tutor, Ms Toh, to clarify any doubts you may have. Also, Ms Toh is by far one of the best mathematics teachers I have ever had as she is able to explain clearly the concept being taught/the thought process in solving the question at hand and as such, I would definitely recommend my friends to join The Maths Cafe. (:
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Scoring Well, Studying Smart

Miss Toh! I really want to thank you for EVERYTHING. No matter how hard I studied last time, I always didn’t manage to score as well as I wanted to. The Maths Cafe questions are tough and prepared me well for my school exams. The colourful summary is concise and extremely handy. What you taught us was not just how to score in Maths, but rather how to study smart. Thank you for all the encouragement, tips and support till the end. I will definitely recommend The Maths Cafe to everyone I know.
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I would definitely recommend The Maths Cafe to my peers. Miss Toh has really helped me pull up my Additional Mathematics grades from a D7 during the Mid-year exams to an A2 in O Levels. She is willing to put in extra time and effort to make sure you understand everything clearly. The notes are amazing as well and have definitely helped me understand topics better. I would like to take this time to thank Miss Toh for being such an amazing mentor.
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