Linear motion (3.1.2)
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Your respective data booklet contains four kinematic equations of motion for objects moving with constant acceleration. It is useful to note that a fifth equation (5) can be memorised to save time in your exams:
The variables in these equations are:
- time ,
- initial velocity ,
- final velocity ,
- acceleration , and
- displacement .
All quantities apart from time are vectors, meaning they can be positive or negative depending on direction.
The choice of equation depends on the given variables for a problem. For instance, if the final velocity is not given or needed, equation (4) is often the most useful.
If an object moves right with a positive velocity, a negative velocity indicates motion to the left. You can choose which direction to take as positive as long as you remain consistent.
Similarly, positive displacement means movement in the chosen positive direction, while negative displacement means movement in the opposite direction.
Acceleration is:
- positive when it causes the magnitude of velocity to increase in the positive direction or decrease in the negative direction.
- negative when it causes the magnitude of velocity to decrease in the positive direction or increase in the negative direction.
When an object falls in a uniform gravitational field, the constant acceleration is determined by gravity. Near Earth’s surface, this acceleration is commonly denoted as This means that, neglecting air resistance, the velocity of a freely falling object increases by about every second.
The kinematic equations are only valid when air resistance is negligible. In this ideal case, all objects fall toward Earth at the same rate. In reality, factors such as an object’s mass and surface area influence air resistance, which in turn affects its motion.
The true value of can vary slightly depending on factors such as altitude and geographical location.
Sometimes, it may seem like a problem requiring the equations of motion to solve does not give you enough information to answer correctly. However, you often have to interpret some phrases in the question to get all the information. Here are some common phrases to watch out for:
- “Starts from rest” – usually means that
- “At maximum height” – the velocity at maximum height is zero.
- “Falling due to gravity” – acceleration is
- “Speed” – if the question asks to calculate speed, then you should ignore the sign of the velocity.
Question walkthrough
Maximum Height of a Ball Thrown Upwards
Finds the maximum height reached by a ball thrown vertically upwards, using v² = u² + 2as with the final velocity equal to zero.
Question walkthrough
Displacement and Acceleration from Changing Velocity
Uses SUVAT equations to find the displacement and acceleration of a car whose velocity changes from positive to negative over a given time.
Question walkthrough
Time of Flight for a Ball Thrown Upwards
Finds the total time a ball is in the air after being thrown upwards and caught at the same height, using a SUVAT equation with s = 0.
Question walkthrough
Finding Acceleration Graphically from s and t²
Describes an experimental method for finding a trolley's acceleration down a ramp by plotting displacement against time squared and using the gradient.
An experiment to measure involving a steel ball-bearing, an electromagnet, a trap-door and a timer is shown below.
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An electromagnet releases the steel ball, which triggers a timer to start. When the ball lands on the trapdoor, a trigger stops the timer and records the time taken.
The distance the ball falls can be measured with a ruler. The relevant equation of motion is:
We know that the ball starts from rest so and Therefore:
If the ball bearing is small, there will be some air resistance, but it should be small. The influence of air resistance can be minimised by ensuring the final velocity is not too high or by using a vacuum chamber.
Question walkthrough
Finding g Graphically from a Pendulum
Describes an experimental method for finding g by measuring the time period of a pendulum for different lengths and plotting T² against length.
When a driver needs to stop a vehicle, it does not happen instantly. The total stopping distance is made up of two parts: the thinking distance and the braking distance:
- Thinking distance – The distance a vehicle travels in the time the driver takes to react.
- Braking distance – The distance a vehicle travels after the brakes are applied until it stops.
- Stopping distance – The total distance required for a vehicle to stop.
Thinking distance depends on how quickly a driver reacts and how fast the vehicle is moving. It can be calculated using a simple formula and can be influenced by several key factors.
Factors Affecting Thinking Distance:
- Reaction time – Affected by:
- Tiredness
- Alcohol or drugs
- Distractions (e.g., mobile phone use)
- Age and experience
- Speed of the vehicle – Higher speed increases thinking distance since the car covers more ground per second.
A driver can engage the brakes to apply a force opposite to the vehicle’s motion, reducing its speed and therefore its kinetic energy.
The work done by the brakes is equal to the initial kinetic energy of the vehicle.
Therefore, braking distance
Significant factors affecting braking distance:
- Speed – Doubling the speed quadruples braking distance.
- Tyre condition – Worn or under-inflated tyres reduce friction, increasing braking distance.
- Brake condition – Worn brakes reduce braking efficiency.
- Road conditions – Wet, icy, or gravel roads increase stopping distances.
- Mass of the vehicle – Heavier vehicles have more kinetic energy, requiring a longer braking distance.
Thinking Distance (Reaction Time Phase) – Velocity remains constant. The area under the graph represents the distance the vehicle travels while the driver reacts before applying the brakes.
Braking Distance (Deceleration Phase) – Velocity decreases linearly to zero. It represents the distance travelled while the vehicle is slowing down due to braking. The area of this triangle corresponds to the braking distance.
The total stopping distance is the sum of the thinking distance and the braking distance, equal to the total area under the graph.
Question walkthrough
Factors Affecting Braking Distance
Explains how factors like initial speed, brake and tyre condition, road conditions, and vehicle mass affect braking distance using the work-energy relationship Fd = ½mv².