Equations of motion and graphs (Topic 2A)
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Displacement is a vector quantity characterised by its magnitude and direction. The net displacement of an object depends solely on its initial and final positions. Any intermediary stops are irrelevant. The magnitude of a displacement vector is equal to the length of the segment joining the initial and final positions.
Distance is a scalar quantity that is fully described by its magnitude. It represents the length of the path an object follows. Distance and displacement are not identical quantities. A person who walks on a circular path and returns to their initial position has a displacement of zero, but has covered a distance equal to the circumference of the circle.
It is important to note that distance travelled and displacement are equal only when an object is moving in a straight line in the positive direction.
Average velocity is a vector quantity defined as the change in the displacement of an object over time:
The direction of the average velocity vector is always the same as the direction of the displacement vector.
Average speed is a scalar quantity defined as the distance covered over time:
Average speed and average velocity are expressed in the SI system of units by metres per second
It is important to note that the magnitudes of average velocity and average speed are not the same. They are equal only when an object is moving in a straight line and in the positive direction.
Instantaneous speed, often simply referred to as speed, is how fast an object is moving at a specific moment. By looking at the speedometer of a moving car, you can read its speed. Speed is a scalar quantity with magnitude but no direction.
Instantaneous velocity, or simply velocity, is used when speed is specified in a given direction. For example, a car moving east at has a speed of and a velocity of due east. The magnitude of instantaneous velocity of a moving object is always equal to its speed.
Speed and velocity are expressed in the SI system of units by metres per second
Acceleration is a vector quantity defined as the change in velocity over time. It indicates how quickly an object’s velocity changes, whether in magnitude, direction, or both:
Acceleration is expressed in the SI system of units by metres per second squared
A common mistake is interpreting the sign of acceleration. Many assume that a negative acceleration indicates that an object is slowing down, while a positive acceleration indicates that the object is speeding up. However, this interpretation depends on the chosen positive direction. For example, a car could be speeding up in the negative direction.
To resolve this, multiply the velocity of the moving object by its acceleration. If you get a negative number, the object is slowing down; if you get a positive number, the object is speeding up. The car shown below is speeding up because the product of its velocity and acceleration is positive.
Question walkthrough
Distance vs Displacement on a Track
Compares distance and displacement, then average speed and average velocity, for an athlete running three sides of a rectangular track.
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.
To analyse motion and collisions experimentally, various techniques and apparatus can be used to collect precise data on velocity, acceleration, and momentum.
Common Apparatus:
- Trolleys & Air-Track Gliders – Minimise friction for accurate motion studies.
- Ticker Timers – Produce dot traces on tape to measure speed and acceleration.
- Light Gates & Data Loggers – Record time and velocity electronically for high accuracy.
- Video Analysis – Captures motion for frame-by-frame study of velocity and collisions.
Choosing the appropriate method depends on the required precision and the type of motion being investigated.
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.
A displacement–time graph (x–t graph), illustrates how an object’s displacement varies with time. Displacement is plotted on the vertical axis, with time along the horizontal axis.
If the line on the x–t graph is moving away from the t-axis with respect to time, then the object is moving away from the origin. Similarly, if the line on the x–t graph is moving towards the t-axis with respect to time, then the object is moving towards the origin.
When an object is moving in the positive direction and away from the origin, a curve that is concave down, as in graph (B), indicates the object is slowing down, and a curve that is concave up, as in (C), indicates that the object is speeding up. A straight oblique line indicates that the object is moving at a constant speed, as in graph (D), or at rest, as shown in graph (A).
A velocity–time graph (v–t graph) shows how the velocity of an object varies with time. Velocity is plotted on the vertical axis with time along the horizontal axis.
If the line on the v–t graph is moving away from the t-axis with respect to time, then the object is speeding up. Similarly, if the line on the v–t graph is moving towards the t-axis with respect to time, then the object is slowing down. This is shown in graphs (B) and (C), respectively.
A horizontal line on the v–t graph, where indicates that the object is moving at a constant speed, as in graph (D); if the horizontal line is on the t-axis, then the object is stationary, as in graph (A).
When the line on the v–t graph is above the t-axis, the object is moving in the positive direction and when the line is below the t-axis, this indicates that the object is moving in the negative direction.
A speed–time graph (v–t graph) is similar to a velocity-time graph in all its aspects except that the former does not indicate the direction of motion. Speed is plotted on the vertical axis and time along the horizontal axis. While a velocity–time graph can show lines that are below the t-axis, a speed–time graph does not.
Consider the following scenario:
- A car has an initial velocity of
- The car starts to slow down at a constant rate until it comes to a momentary stop.
- The car then starts to speed up in the negative direction until it reaches a velocity of
The velocity–time graph of the car is shown in graph (A), and the speed–time graph of the same car is shown in graph (B). The negative values in the velocity–time graph are reflected as positive values in the speed–time graph.
An acceleration–time graph (a–t graph) shows the rate of change of the velocity with time. Acceleration is plotted on the vertical axis, while time is along the horizontal axis:
- A horizontal line on the a–t graph that aligns with the t-axis indicates that the object is not accelerating, as shown in graph (A).
- A horizontal line above or below the t-axis indicates that the object is moving at a constant acceleration, as shown in graph (B).
Data loggers are powerful instruments that are used to measure and analyse the motion of objects. A data logger is an electronic device that records data over time. It consists of sensors that can be used to record:
- position,
- speed,
- acceleration
Using data from data loggers, we can draw the following graphs:
- displacement–time
- velocity–time
- acceleration–time
Then analyse motion and get valuable insights.
Data loggers enhance motion analysis by offering precise, reliable, real-time measurements and immediate feedback, enabling quick adjustments. Their software also allows for real-time visualisation of motion graphs.
The gradient of the tangent to any point on the displacement–time graph is equal to the instantaneous velocity at that point:
| Gradient / velocity | Meaning |
|---|---|
| Positive gradient on displacement–time graph | Positive velocity |
| Negative gradient on displacement–time graph | Negative velocity |
| Positive velocity | Object moving in the positive direction |
| Negative velocity | Object moving in the negative direction |
The numerical value of the gradient can be determined by drawing a right triangle using any two points on the tangent. The vertical side of the triangle is called the rise, and the horizontal side is called the run.
The gradient can be calculated using:
The variation of the tangent’s gradient on a displacement–time graph can indicate whether an object is speeding up, slowing down, moving at a constant speed, or at rest.
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- Graph (A): A tangent that gets steeper with time indicates that the object is speeding up.
- Graph (B): If the tangent gets less steep with time, the object is slowing down.
- Graph (C): A horizontal tangent whose gradient is equal to zero, which indicates that the object is at rest.
- Graph (D): A tangent whose gradient does not vary with time (i.e. constant) signifies that the object is moving at a constant speed.
The gradient of the tangent to any point on the velocity–time graph is equal to the acceleration at that point:
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| Gradient / acceleration | Meaning |
|---|---|
| Positive gradient on velocity–time graph | Positive acceleration |
| Negative gradient on velocity–time graph | Negative acceleration |
| Positive acceleration | Object’s velocity is increasing with time |
| Negative acceleration | Object’s velocity is decreasing with time |
The variation of the tangent’s gradient on a velocity–time graph can indicate whether the acceleration of an object is uniform, non-uniform, or equal to zero:
- Graph (A): A gradient that changes with time indicates that the acceleration is non-uniform.
- Graph (B): A horizontal tangent indicates that the object’s acceleration is zero.
- Graph (C): Showcases a tangent where the gradient does not vary with time, signifying that the object is accelerating uniformly.
The area under the velocity–time graph is equal to the displacement of the object. When the curve on the velocity time graph is above the t-axis, then the area under it is positive. If the curve is below the t-axis, the area under the curve is negative.
The best way to determine the area under the graph is to divide it into regular geometric shapes of known area, such as rectangles, triangles, and trapezoids, and then sum the individual areas.
In a nonlinear velocity–time graph, you can get a rough estimate of the area under the curve by dividing the area into small trapezoids, triangles, and rectangles. The smaller the divisions, the more accurate the estimation will be.
Using trapezoids is particularly effective because it accounts for the curvature of the graph better than rectangles. Do not forget that areas under the t-axis have negative values.
The area under the graph is the sum of the individual areas of the shapes. For the graph above:
Question walkthrough
Finding Displacement from a Velocity-Time Graph
Splits a velocity-time graph into a triangle, trapezoid, and triangle to calculate the total displacement as the sum of the three areas.