Notes for Linear Motion Test

Summary

The upcoming CAP is on linear motion, comparatively the easiest type we’ll be having throughout. Though most of it is equations, there’s still some concepts that are important to remember. (This is straight from the checklist, link to that is in this notice here, but) The general topics that will be covered are:

1. Free Body and Vector Diagrams

Free body diagrams (FBD) indicate the force acting on an object/body. These forces are indicated with arrows in the direction the force is acting. Usually, the length of the arrow is to scale in terms of the magnitude of the force they are representing (for example, if the magnitude of one force is 4N and the magnitude of another is 2N, the length of the arrow representing the former would be twice of that of the latter; there is no need for this if the value is not given). They are used for finding the net force acting on an object. All arrows drawn in a FBD must be labelled, and extraneous ones are heavily discouraged if they are not important to calculating the net force acting on an object. These labels include both the magnitude of the force and units (which is Newtons) if you are meant to quantify them, or if you are not given these, then the type of force being represented. If you are making new arrows to find net force, then ensure the line for the arrow is dotted. To account for gravity, gravitational force, weight force, gravity or $f_g$/$f_w$ is fine as the label without a mass value. If there is a mass value given, you will need to calculate the weight force acting on the object and label the arrow with it. You will also need a reference for the directions, where the stock standard is an arrow pointing upwards labelled North/N. Below is an example of a FBD.

Vector diagrams on the other hand represent the vector quantities applied to a certain object, and, similar to FBD. Arrows are also used to indicate vectors, but the arrow heads can be anywhere on the line, and they do not have to connect to the object in certain cases (e.g. calculating net velocity). In general, vector diagrams are practically the same, in the sense that the same concepts apply from free body diagrams to it, just that vector diagrams show vector quantities instead. Below is an example of a vector diagram.

2. Scalar and Vector Quantities

Pretty much the most straightforward concept in physics, but many silly mistakes are made in this area. Scalar quantities are quantities that only involve magnitude or size of a measurement. Examples of scalar quantities are temperature (C˚), distance (m) and speed ($m s^{-1}$; I know that it’s better for the units to be separated but for the life of me I cannot figure out Latex). Vector quantities are quantities that involve both a magnitude and direction for a measurement. Examples are velocity ($m s^{-1}$ North), acceleration ($m s^{-2}$ North) and force (N/$kg m s^-2$ (it’s kilogram-metres per second per second) North).

Scalar quantities aren’t going to be much of a problem, so it’s highly likely there’ll be less of it in the test, just that you need to remember that a direction is not needed; rather, you may be penalised if you do put one. Vector quantities, however, introduce negative quantities (as you can be going negative North (which is, of course, South)). This must be accounted for in answering questions involving vector quantities, so it is best to create a reference point and label a positive and negative from there. For example, if you had a question with a force acting on the object to the East, and another force acting to the West, then label one as the positive side, then subtract the other force from it (since they are directly opposite). Note that this only is useful for directly opposite things, it’s better to use a vector diagram and/or some trigonometry when there are vector quantities that are not in the same direction but are not completely opposite (breaking into component form can also be used with the former method, but it might take more time depending on what values are given).

3. Equations of Motion

Another quite straightfoward concept, all of the formulas are in the data booklet that is provided to you. If you do not know what the pronumerals are, here is the list:

s = displacement

u = initial velocity

v = final velocity

a = acceleration

t = time (rather, change in time or $∆t$)

Note that s is also distance and v is also speed, so be careful when using them. The negative and positive (See 2. Scalar and Vector Quantities) method also follows here, so make sure to account for it.

4. Laws of Motion

Some of these are slightly misunderstood, so here are widely accepted textbook definitions.

  1. An object will continue to stay at rest or at a state of uniform motion in the same direction unless acted upon by an external unbalanced force.
  2. The acceleration of an object is directly proportional to the force applied on said object, and is also inversely proportional to the mass of the object (F = ma is acceptable in some cases).
  3. Every reaction has an equal (same magnitude) and opposite (in the opposite direction) reaction. (This does NOT mean that if you kick a ball, the ball is applying the same force against your foot and therefore does not move. The ball has a significantly smaller mass compared to your foot, and hence, your body (surprise surprise they are connected to each other), so the ball will obviously accelerate more than your foot will)

5. Momentum and Impulse

Probably the two that will confuse some people. Momentum is a conserved vector quantity that is a measure of the product of an object’s mass and velocity, measured in $kg m s^{-1}$. Impulse is a change in momentum but there is something conceptually different with the two, which will be discussed later.

(Sum of) Momenta do not change unless an object has another force acting on it (because a force is that which causes a mass to accelerate, changing the velocity, therefore changing momentum), so even in collisions, separations and coalescences, the sum of momenta of the object remain the same. This is called the Law of Conservation of Momentum.

The formula for collisions (opposite directions) is $p_1-p_2=p_{final}$

The formula for separations is $p_{total}=p_1+p_2$

The formula for coalescences is $p_1+p_2=p_{final}$

However, due to forces acting in our daily lives, momentum can be lost. This helps in some situations, for example, when a cricketer follows the motion of a cricket ball when catching it. If they caught the ball without moving their hand backwards, the ball is slowed down from its current velocity to zero velocity, and therefore, zero momentum. This huge change in momentum can easily injure the cricketer’s hand since it is across a short period of time. When they follow the ball’s path before catching it, the change in momentum happens across a larger period of time, therefore the average force acting on the cricketer’s hand will reduce, causing less damage.

The same concept is also applied to crumple zones and safety barriers (e.g. bull bars). A crash happens across an extremely short period of time, and change in momentum across a short period of time, as we know, has a higher amount of damage. Crumple zones increase the amount of time that the crash takes place, as the car crushes more easily and rather than the full brunt of the force affecting the passengers, the momentum is lost across the crumple zone (normal force), so it will have significantly less damage done to the passengers. Bull bars also have a similar effect, as well as the fact it distances the vehicle from another so more of the damage is done to it rather than the vehicle. Helmets, seatbelts and airbags too have this effect, but in differing qualities. Helmets, despite having essentially the same effect as bull bars on your head except that it covers the entirety of it, do not prevent internal concussion due to significant trauma. Seatbelts are also the same, causing whiplash in some car crashes, but are comparatively better than nothing. Airbags are also in on this increasing time scheme, but they significantly reduce the chance for a driver and passengers to have significant concussion since they envelop them at all sides, applying normal force that counteracts the force that causes drivers and such to rock back and forth.

While impulse is known as a change in momentum, their units differ due to the formula being different ($N s$ is for impulse, $kg m s^-1$ is for change in momentum). Also, momentum divided by change in time is NOT force, impulse divided by change in time is (because the gradient of a momentum to time graph is force, but a gradient is calculated through the change of the values). Similarly, in analysing graphs, impulse can be calculated by finding the area under a force to time graph.

6. Energy, Work and Power

These may also appear daunting to some, but the main issue is in definitions, the rest is fairly simple. Energy is the ability to do work, and is a conserved scalar quantity. It is measured in Joules (J), which, in result, is $kg m^2 s^{-2}$. Energy can be split into two different categories: kinetic energy and potential energy. Kinetic energy is the energy possessed by an object from/due to its motion. Potential energy is energy possessed by an object due to its position in a force field, and hence, can be thought as stored energy. The formulae for these two are in your data booklet.

Work is known as the change in energy (though energy is conserved and cannot be created nor destroyed, energy can be converted, so work on focuses on a specific type of energy) and is also measured in Joules (J). Work is also done when a force’s point of application is moved in the direction of the said force (hence, work is the product of force and displacement). No work is done if an object is at rest or remaining at a constant velocity. However, lifting an object, accelerating and decelerating it result in work being done (as all involve changes in velocity, or acceleration in some way). Objects with potential energy too have the ability to do work due to their position.

Another concept is elastic and inelastic collisions. Elastic collisions between objects involve no change in the total kinetic energy of the objects, and is therefore conserved. On the other hand, in inelastic collisions, kinetic energy of the objects can be converted to another type, therefore the kinetic energy of the objects will not be the same. There usually does not exist completely elastic collisions except for that of particles in the air right now (kinetic theory of matter), but for the sake of the test, there could be for all we know.

Power is the rate of change of energy (change in energy divided by change in time). Also can be the rate of work done, and is measured in Watts (W) or $J s^{-1}$ (Joules per second; best to take it into account when analysing graphs).

Other things to cover

The main thing to consider that I haven’t mentioned yet is analysing graphs. Finding the area under the curve and the gradient usually isn’t too hard, but for non-linear graphs, just estimate by using two different points or tangents to certain points. You are not meant to do calculus for this test, but even if you do, there’s no guarantee you’ll get marks.

Definitions

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