A Quick Guide to Motion and Kinematics

[jamonwindeyer]

Hello everyone! Welcome to the first in a series of guides which will give you some quick revision of the core topics of HSC Physics. These guides will aim to go over the content very quickly, so reading them should jog your memory of a whole lot of content (a very good thing with exams just over 3 months away). This, combined with some exam question examples, will hopefully prove an awesome resource for your study for Trials and the HSC.

I wanted to write this first guide to revise some content which is covered more deeply in Year 11. Kinematics, and analysis of motion, is something which is sort of hidden away in HSC Physics. But it is an essential part of the course, since pretty much all First Year Physics courses at uni are heavily based on kinematics, and will always be the basis of at least a decent amount of marks in the HSC exam. This guide will revise the key concepts from Preliminary Physics, and address what you need to know in HSC, to understand why things move the way they do.

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Okay, so let's quickly revise the basics. We understand that there are units of distance, and units of time. Speed is the distance travelled per unit time (i.e., speed is a measure of the rate of change of distance). Similarly, acceleration is the measure of the rate of change of speed). While you don't have to do any calculus to analyse these relationships, it is important to at least know of them. It is how all the formulas are derived (again, something you don't have to do).

But why do things move the way they do? This is where forces come in. Forces are pushing, pulling or twists motions which cause the acceleration of an object. Isaac Newton formulated the infamous three laws:

1- An object will not experience an acceleration unless a non zero net external force acts on the object.
2- \(F=ma\)
3- For every action force, there is an equal and opposite reaction force.

Forces are an example of a vector quantity . Vectors are simply quantities where you must consider direction as well as magnitude. Going 3km north is clearly different to going 3km south, for example. Quantities where no direction is involved are called scalars.

In Year 11 you also are introduced to two other ideas. First, the idea of energy. Energy is the the capacity to do work , and the energy associated with motion is called kinetic energy. You are also taught about momentum. This is a property of a moving mass similar to inertia, and is defined as . Both of these are conservative quantities . That is, neither energy nor momentum can be created or destroyed. They can only be transferred, or in the case of energy, transformed.

These are the basic ideas you should have some familiarity with from Year 11 (there are more, but they have less relevance). They don't come into questions specifically, but they are a vital stepping stone to accessing the questions in the HSC.

In the HSC, you focus on the idea of projectiles. A projectile can be defined in a number of ways, but in the HSC, think of a projectile only in terms of its motion.

We consider the motion of projectiles in terms of separate x and y axis components. The motion of all objects can be considered this way, and using trigonometry, it is actually fairly easy to do. Consider the example below, which shows how a velocity can be resolved into vertical and horizontal components using simple right angled trigonometry.



A projectile is acted on by a single force; gravity, in the vertical direction. Since we know that forces are required to produce acceleration, this means that only the vertical motion of the object experiences an acceleration, in this case equal to \(g\), or \(9.8ms^{-1}\) (on earth). The horizontal component experiences no acceleration; thus, the horizontal motion remains constant. This is what creates the parabolic path of a projectile through the air. Galileo proved all of this mathematically based on observations made using an incline plain.

Putting all of this together, we get the following formulae for analysing projectile motion. For the x axis, or any scenario with constant velocity and no acceleration;



And for the y axis, or any scenario with a constant acceleration;



Where v is instantaneous velocity, u is initial velocity, s is displacement, t is time elapsed, and a is acceleration.

Questions concerning projectiles usually involve choosing these formulas carefully to predict a value such as range, time of flight, time of impact, etc. When tackling these questions, it is important to remember a few key facts:

• The vertical velocity of a projectile at its highest point (i.e., the peak of motion), is zero. This can be a useful start for many projectile questions.
• The time taken to reach the maximum height is half that taken to return to the initial height
• The maximum range of a projectile is achieved when it is launched at an angle of 45/degree
• Any x position for impact with the ground (or initial height) less than the maximum range can be achieved with two different launch angles (you don't need to prove this, and it rarely comes up, but good to keep in mind)

Most HSC exams will have a projectile question worth at least 4 marks. The last few papers have done slightly stranger questions, referring to practicals or asking for graphs. This could mean a new trend has started. Or, it could guarantee that we will see a return to tradition. Either way, questions like this are much more common:



There are numerous ways to approach this question. First, however, we should resolve the velocity into horizontal and vertical components. We wish to find the initial velocity, \(V\).

The horizontal velocity can be expressed as \(V\cos{60}\), and the vertical, \(V\sin{60}\). This can be easily seen by drawing a triangle similar to that above. Don't round anything yet, I personally leave the trig ratios there, but you can also put it in surd form if you like. Just don't round until the end!

Next, consider the range. We know that it will travel 45 metres horizontally. Therefore, since the horizontal velocity is constant, we can use the simple formula:



We can substitute this value in when considering vertical motion, at a height of 35m, to solve for V.



We take the positive value of the square root in that last line, we are asked for a magnitude (no direction, so no sign necessary, nor does it make sense to make it negative).

Something else which can pop up is quadratics, particularly for solving for time of flight. Be careful when this occurs. Time can obviously not take a negative value. In general, when solving quadratics, only one answer will be of use to you; think carefully about which. Also be careful of differing values for g (they will always give you these in the question).

The other thing which is essential to understand in the HSC is uniform circular motion. Uniform circular motion occurs when a force is applied perpendicular to the instantaneous velocity of an object. This results in circular motion, with a centripetal force applied towards the centre. In the HSC, this centripetal force is applied by gravitational force, but we'll look at that in another guide. For now, just understand why uniform circular motion occurs, and remember these formulae for uniform circular motion in general:



Kinematics doesn't play as large a role as other areas of Physics in the HSC, but given how important it is for tertiary study, and the fact that a projectile question of some kind always appears, it is definitely worth looking over. Practice some questions, be sure you understand how it works!

That's all for this guide. Stay tuned for more, and remember to register for an account and questions below! This is my study area at Uni, and helping you guys helps me revise as well, so fire away!

A GUIDE BY JAMON WINDEYER