Kinematics

What you'll learn

How to describe motion using displacement, speed, velocity and acceleration.
How to tell the difference between scalar and vector quantities.
How to extract motion information from displacement–time and velocity–time graphs.
How to use gradients and areas to calculate velocity, acceleration and displacement.

1. Describing motion clearly

Kinematics is the study of motion without worrying about what caused the motion. In this section, you are mainly asking: where is the object, how fast is it moving, and how is its motion changing?

Scalars and vectors

A scalar quantity has magnitude only. Magnitude means size. Examples include distance, speed, time and mass.

A vector quantity has both magnitude and direction. Examples include displacement, velocity, acceleration and force.

Definition

Scalar and vector

A scalar has magnitude only. A vector has magnitude and direction, so signs such as positive and negative can represent direction in one-dimensional motion.

Distance and displacement

Distance is the total length of the path travelled. It does not include direction, so it is a scalar.

Displacement is the straight-line change in position from the starting point to the finishing point, in a stated direction. It is a vector.

For example, if you walk 3 m east and then 3 m west, your distance travelled is 6 m, but your displacement is 0 m because you ended where you started.

Common Mistake

Distance is not displacement

If an object turns around, the distance travelled keeps increasing, but the displacement can decrease or even return to zero.

Speed and velocity

Speed is the rate of change of distance. It is a scalar, measured in metres per second, m s $^{-1}$ .

The average speed over a journey is:

taken\text{average speed} = \frac{\text{total distance travelled}}{\text{time taken}}

Velocity is the rate of change of displacement. It is a vector, also measured in metres per second, m s $^{-1}$ . In one-dimensional motion, a positive velocity means motion in the chosen positive direction; a negative velocity means motion in the opposite direction.

Definition

Instantaneous speed

Instantaneous speed is the speed of an object at a particular instant in time, like the reading on a speedometer.

Acceleration

Acceleration is the rate of change of velocity. It is a vector, measured in metres per second squared, m s $^{-2}$ .

For a change in velocity $Δv\Delta v$ over a time interval $Δt\Delta t$ :

\frac{\Delta v}{\Delta t}

If acceleration is in the same direction as velocity, the object speeds up. If acceleration is in the opposite direction to velocity, the object slows down.

Key Idea

Acceleration is about velocity, not just speed

An object can be accelerating because its speed changes, its direction changes, or both. In this topic you mostly deal with one-dimensional motion, so direction is usually shown using positive and negative signs.

Example

Comparing average speed, average velocity and acceleration

A cyclist travels 120 m east in 20 s, then 40 m west in the next 10 s. Their velocity changes from 8.0 m s $^{-1}$ east to 4.0 m s $^{-1}$ west during those final 10 s. Take east as positive.

The total distance travelled is $\text{ m}$ , and the total time is $\text{ s}$ , so the average speed is:
$s−1\frac{160 \text{ m}}{30 \text{ s}} = 5.3 \text{ m s}^{-1}$
The final displacement is $\text{ m}$ east, so the average velocity is:
$east\frac{80 \text{ m}}{30 \text{ s}} = 2.7 \text{ m s}^{-1} \text{ east}$
During the final 10 s, the velocity changes from $\text{ m s}^{-1}$ to $\text{ m s}^{-1}$ , so:
$\frac{-4.0 - 8.0}{10} = -1.2 \text{ m s}^{-2}$
The negative acceleration means the acceleration is westwards.

2. Graphs as motion stories

Graphs are a compact way to represent motion. Always check:

what is on the vertical axis,
what is on the horizontal axis,
the units,
whether the graph is showing displacement, speed, velocity or acceleration.

A data logger, light gate, motion sensor or video-analysis software can collect motion data automatically. This is useful because it gives many data points, making gradients and areas more reliable than if you only had a few manual readings.

Displacement, speed, velocity and acceleration graphs

A displacement–time graph shows how position changes with time.

A speed–time graph shows how the magnitude of speed changes with time. Speed cannot be negative.

A velocity–time graph shows how velocity changes with time. Velocity can be positive, zero or negative.

An acceleration–time graph shows how acceleration changes with time.

Tip

Read the axis label first

A graph labelled speed–time and a graph labelled velocity–time are not the same. Velocity can go below the time axis; speed cannot.

3. Displacement–time graphs

On a displacement–time graph, the gradient tells you the velocity.

graph\text{velocity} = \text{gradient of a displacement–time graph}

For a straight-line section:

\frac{\Delta s}{\Delta t}

where $Δs\Delta s$ is the change in displacement and $Δt\Delta t$ is the change in time.

If the graph is curved, the velocity is changing. The instantaneous velocity at one point is found from the gradient of the tangent to the curve at that point.

Displacement-time graph showing average velocity from a chord and instantaneous velocity from a tangent

What the shape tells you

A steeper positive gradient means a greater positive velocity.
A horizontal line means zero velocity because displacement is not changing.
A negative gradient means negative velocity, so the object is moving in the opposite direction.
A curve means changing velocity.

Example

Finding velocity from a displacement–time graph

A straight section of a displacement–time graph goes from 2.0 m at 1.0 s to 14.0 m at 5.0 s. Find the velocity during this section.

Use the gradient because this is a displacement–time graph:
$\frac{\Delta s}{\Delta t}$
Calculate the change in displacement and the change in time:
$m\Delta s = 14.0 - 2.0 = 12.0 \text{ m}$ $s\Delta t = 5.0 - 1.0 = 4.0 \text{ s}$
Substitute into the gradient calculation:
$\frac{12.0 \text{ m}}{4.0 \text{ s}} = 3.0 \text{ m s}^{-1}$

Common Mistake

Using the graph height instead of the gradient

On a displacement–time graph, the vertical value is displacement, not velocity. Velocity comes from the gradient.

4. Velocity–time graphs

On a velocity–time graph, the gradient tells you the acceleration.

graph\text{acceleration} = \text{gradient of a velocity–time graph}

For a straight-line section:

\frac{\Delta v}{\Delta t}

The area under a velocity–time graph gives displacement.

graph\text{displacement} = \text{area under a velocity–time graph}

This works because the units of area are:

s−1×s=m\text{m s}^{-1} \times \text{s} = \text{m}

Velocity-time graph showing acceleration from gradient and displacement from area under the graph

Positive and negative areas

If the velocity is above the time axis, the area is positive displacement. If the velocity is below the time axis, the area is negative displacement.

If you want distance travelled, add the magnitudes of the areas. If you want displacement, include the signs.

Key Idea

Gradient and area do different jobs

On a velocity–time graph, the gradient gives acceleration, while the area under the graph gives displacement.

Example

Using a velocity–time graph

A car’s velocity increases uniformly from 0 m s $^{-1}$ to 12 m s $^{-1}$ in 4.0 s, then stays at 12 m s $^{-1}$ for 6.0 s. Find the acceleration during the first 4.0 s and the displacement during the full 10.0 s.

Use the gradient of the first section for acceleration:
$\frac{12 - 0}{4.0} = 3.0 \text{ m s}^{-2}$
Split the area under the graph into a triangle and a rectangle. The triangle has base 4.0 s and height 12 m s $^{-1}$ :
$m\text{triangle area} = \frac{1}{2} \times 4.0 \text{ s} \times 12 \text{ m s}^{-1} = 24 \text{ m}$
The rectangle has width 6.0 s and height 12 m s $^{-1}$ :
$m\text{rectangle area} = 6.0 \text{ s} \times 12 \text{ m s}^{-1} = 72 \text{ m}$
Add the areas to find the total displacement:
$\text{ m} + 72 \text{ m} = 96 \text{ m}$

5. Estimating area under a non-linear velocity–time graph

If a velocity–time graph is curved, the area may not be a simple triangle or rectangle. You can estimate the displacement by:

counting squares under the curve,
splitting the area into trapezia,
using narrower strips for a better estimate.

For a trapezium with parallel sides $a$ and $b$ , separated by width $h$ :

area=12(a+b)h\text{area} = \frac{1}{2}(a+b)h

On a velocity–time graph, the parallel sides are velocity values, and the width is the time interval.

Example

Estimating displacement with trapezia

A velocity–time graph has these readings: at 0 s, 0 m s $^{-1}$ ; at 2.0 s, 5.0 m s $^{-1}$ ; at 4.0 s, 8.0 m s $^{-1}$ ; at 6.0 s, 9.0 m s $^{-1}$ . Estimate the displacement from 0 s to 6.0 s.

Use three trapezia, each with width 2.0 s, because the velocity readings are separated by equal 2.0 s intervals.
Calculate each trapezium area:
$m\frac{1}{2}(0 + 5.0)(2.0) = 5.0 \text{ m}$ $m\frac{1}{2}(5.0 + 8.0)(2.0) = 13 \text{ m}$ $m\frac{1}{2}(8.0 + 9.0)(2.0) = 17 \text{ m}$
Add the estimated areas:
$\text{ m} + 13 \text{ m} + 17 \text{ m} = 35 \text{ m}$

Common Mistake

Area gives displacement, not always distance

If part of a velocity–time graph is below the time axis, its area is negative. For total distance travelled, add the sizes of the positive and negative areas instead.

6. Acceleration–time graphs

An acceleration–time graph shows how acceleration varies with time. A horizontal line means constant acceleration. A line on the time axis means zero acceleration, so the velocity is constant.

The area under an acceleration–time graph gives the change in velocity, because:

s−1\text{m s}^{-2} \times \text{s} = \text{m s}^{-1}

You will use the same graph skills: read the axes carefully, use gradients only when the graph type calls for them, and use units to check what your answer represents.

Tip

Use units as a sanity check

If your calculation gives m s $^{-2}$ , it is probably an acceleration. If it gives m, it is probably a displacement or distance. If it gives m s $^{-1}$ , it is probably a speed or velocity.

Exam technique

In the exam

Identify the graph type before doing anything: displacement–time, velocity–time, speed–time or acceleration–time.
For gradients, choose two well-spaced points on the line or tangent, then calculate change in vertical quantity divided by change in horizontal quantity.
For areas under velocity–time graphs, split the shape into rectangles, triangles or trapezia, and keep negative areas negative if displacement is required.

Self review

Check yourself

What is the difference between average speed and average velocity for a journey where the object turns around?
On a displacement–time graph, what does a horizontal line mean?
On a velocity–time graph, how would you find acceleration and displacement?

What you'll learn

How to describe motion using displacement, speed, velocity and acceleration.
How to tell the difference between scalar and vector quantities.
How to extract motion information from displacement–time and velocity–time graphs.
How to use gradients and areas to calculate velocity, acceleration and displacement.

1. Describing motion clearly

Scalars and vectors

A scalar quantity has magnitude only. Magnitude means size. Examples include distance, speed, time and mass.

A vector quantity has both magnitude and direction. Examples include displacement, velocity, acceleration and force.

Definition

Scalar and vector

A scalar has magnitude only. A vector has magnitude and direction, so signs such as positive and negative can represent direction in one-dimensional motion.

Distance and displacement

Distance is the total length of the path travelled. It does not include direction, so it is a scalar.

Displacement is the straight-line change in position from the starting point to the finishing point, in a stated direction. It is a vector.

For example, if you walk 3 m east and then 3 m west, your distance travelled is 6 m, but your displacement is 0 m because you ended where you started.

Common Mistake

Distance is not displacement

If an object turns around, the distance travelled keeps increasing, but the displacement can decrease or even return to zero.

Speed and velocity

Speed is the rate of change of distance. It is a scalar, measured in metres per second, m s $^{-1}$ .

The average speed over a journey is:

taken\text{average speed} = \frac{\text{total distance travelled}}{\text{time taken}}

Definition

Instantaneous speed

Instantaneous speed is the speed of an object at a particular instant in time, like the reading on a speedometer.

Acceleration

Acceleration is the rate of change of velocity. It is a vector, measured in metres per second squared, m s $^{-2}$ .

For a change in velocity $Δv\Delta v$ over a time interval $Δt\Delta t$ :

\frac{\Delta v}{\Delta t}

If acceleration is in the same direction as velocity, the object speeds up. If acceleration is in the opposite direction to velocity, the object slows down.

Key Idea

Acceleration is about velocity, not just speed

Example

Comparing average speed, average velocity and acceleration

A cyclist travels 120 m east in 20 s, then 40 m west in the next 10 s. Their velocity changes from 8.0 m s $^{-1}$ east to 4.0 m s $^{-1}$ west during those final 10 s. Take east as positive.

The total distance travelled is $\text{ m}$ , and the total time is $\text{ s}$ , so the average speed is:
$s−1\frac{160 \text{ m}}{30 \text{ s}} = 5.3 \text{ m s}^{-1}$
The final displacement is $\text{ m}$ east, so the average velocity is:
$east\frac{80 \text{ m}}{30 \text{ s}} = 2.7 \text{ m s}^{-1} \text{ east}$
During the final 10 s, the velocity changes from $\text{ m s}^{-1}$ to $\text{ m s}^{-1}$ , so:
$\frac{-4.0 - 8.0}{10} = -1.2 \text{ m s}^{-2}$
The negative acceleration means the acceleration is westwards.

2. Graphs as motion stories

Graphs are a compact way to represent motion. Always check:

what is on the vertical axis,
what is on the horizontal axis,
the units,
whether the graph is showing displacement, speed, velocity or acceleration.

Displacement, speed, velocity and acceleration graphs

A displacement–time graph shows how position changes with time.

A speed–time graph shows how the magnitude of speed changes with time. Speed cannot be negative.

A velocity–time graph shows how velocity changes with time. Velocity can be positive, zero or negative.

An acceleration–time graph shows how acceleration changes with time.

Tip

Read the axis label first

A graph labelled speed–time and a graph labelled velocity–time are not the same. Velocity can go below the time axis; speed cannot.

3. Displacement–time graphs

On a displacement–time graph, the gradient tells you the velocity.

graph\text{velocity} = \text{gradient of a displacement–time graph}

For a straight-line section:

\frac{\Delta s}{\Delta t}

where $Δs\Delta s$ is the change in displacement and $Δt\Delta t$ is the change in time.

If the graph is curved, the velocity is changing. The instantaneous velocity at one point is found from the gradient of the tangent to the curve at that point.

Displacement-time graph showing average velocity from a chord and instantaneous velocity from a tangent

What the shape tells you

A steeper positive gradient means a greater positive velocity.
A horizontal line means zero velocity because displacement is not changing.
A negative gradient means negative velocity, so the object is moving in the opposite direction.
A curve means changing velocity.

Example

Finding velocity from a displacement–time graph

A straight section of a displacement–time graph goes from 2.0 m at 1.0 s to 14.0 m at 5.0 s. Find the velocity during this section.

Use the gradient because this is a displacement–time graph:
$\frac{\Delta s}{\Delta t}$
Calculate the change in displacement and the change in time:
$m\Delta s = 14.0 - 2.0 = 12.0 \text{ m}$ $s\Delta t = 5.0 - 1.0 = 4.0 \text{ s}$
Substitute into the gradient calculation:
$\frac{12.0 \text{ m}}{4.0 \text{ s}} = 3.0 \text{ m s}^{-1}$

Common Mistake

Using the graph height instead of the gradient

On a displacement–time graph, the vertical value is displacement, not velocity. Velocity comes from the gradient.

4. Velocity–time graphs

On a velocity–time graph, the gradient tells you the acceleration.

graph\text{acceleration} = \text{gradient of a velocity–time graph}

For a straight-line section:

\frac{\Delta v}{\Delta t}

The area under a velocity–time graph gives displacement.

graph\text{displacement} = \text{area under a velocity–time graph}

This works because the units of area are:

s−1×s=m\text{m s}^{-1} \times \text{s} = \text{m}

Velocity-time graph showing acceleration from gradient and displacement from area under the graph

Positive and negative areas

If the velocity is above the time axis, the area is positive displacement. If the velocity is below the time axis, the area is negative displacement.

If you want distance travelled, add the magnitudes of the areas. If you want displacement, include the signs.

Key Idea

Gradient and area do different jobs

On a velocity–time graph, the gradient gives acceleration, while the area under the graph gives displacement.

Example

Using a velocity–time graph

Use the gradient of the first section for acceleration:
$\frac{12 - 0}{4.0} = 3.0 \text{ m s}^{-2}$
Split the area under the graph into a triangle and a rectangle. The triangle has base 4.0 s and height 12 m s $^{-1}$ :
$m\text{triangle area} = \frac{1}{2} \times 4.0 \text{ s} \times 12 \text{ m s}^{-1} = 24 \text{ m}$
The rectangle has width 6.0 s and height 12 m s $^{-1}$ :
$m\text{rectangle area} = 6.0 \text{ s} \times 12 \text{ m s}^{-1} = 72 \text{ m}$
Add the areas to find the total displacement:
$\text{ m} + 72 \text{ m} = 96 \text{ m}$

5. Estimating area under a non-linear velocity–time graph

If a velocity–time graph is curved, the area may not be a simple triangle or rectangle. You can estimate the displacement by:

counting squares under the curve,
splitting the area into trapezia,
using narrower strips for a better estimate.

For a trapezium with parallel sides $a$ and $b$ , separated by width $h$ :

area=12(a+b)h\text{area} = \frac{1}{2}(a+b)h

On a velocity–time graph, the parallel sides are velocity values, and the width is the time interval.

Example

Estimating displacement with trapezia

A velocity–time graph has these readings: at 0 s, 0 m s $^{-1}$ ; at 2.0 s, 5.0 m s $^{-1}$ ; at 4.0 s, 8.0 m s $^{-1}$ ; at 6.0 s, 9.0 m s $^{-1}$ . Estimate the displacement from 0 s to 6.0 s.

Use three trapezia, each with width 2.0 s, because the velocity readings are separated by equal 2.0 s intervals.
Calculate each trapezium area:
$m\frac{1}{2}(0 + 5.0)(2.0) = 5.0 \text{ m}$ $m\frac{1}{2}(5.0 + 8.0)(2.0) = 13 \text{ m}$ $m\frac{1}{2}(8.0 + 9.0)(2.0) = 17 \text{ m}$
Add the estimated areas:
$\text{ m} + 13 \text{ m} + 17 \text{ m} = 35 \text{ m}$

Common Mistake

Area gives displacement, not always distance

If part of a velocity–time graph is below the time axis, its area is negative. For total distance travelled, add the sizes of the positive and negative areas instead.

6. Acceleration–time graphs

An acceleration–time graph shows how acceleration varies with time. A horizontal line means constant acceleration. A line on the time axis means zero acceleration, so the velocity is constant.

The area under an acceleration–time graph gives the change in velocity, because:

s−1\text{m s}^{-2} \times \text{s} = \text{m s}^{-1}

You will use the same graph skills: read the axes carefully, use gradients only when the graph type calls for them, and use units to check what your answer represents.

Tip

Use units as a sanity check

If your calculation gives m s $^{-2}$ , it is probably an acceleration. If it gives m, it is probably a displacement or distance. If it gives m s $^{-1}$ , it is probably a speed or velocity.

Exam technique

In the exam

Identify the graph type before doing anything: displacement–time, velocity–time, speed–time or acceleration–time.
For gradients, choose two well-spaced points on the line or tangent, then calculate change in vertical quantity divided by change in horizontal quantity.
For areas under velocity–time graphs, split the shape into rectangles, triangles or trapezia, and keep negative areas negative if displacement is required.

Self review

Check yourself

What is the difference between average speed and average velocity for a journey where the object turns around?
On a displacement–time graph, what does a horizontal line mean?
On a velocity–time graph, how would you find acceleration and displacement?

What you'll learn

1. Describing motion clearly

Scalars and vectors

Distance and displacement

Speed and velocity

Acceleration

2. Graphs as motion stories

Displacement, speed, velocity and acceleration graphs

3. Displacement–time graphs

What the shape tells you

4. Velocity–time graphs

Positive and negative areas

5. Estimating area under a non-linear velocity–time graph

6. Acceleration–time graphs

Review Kinematics by teaching Genie

Remember key concepts with flashcards

Kinematics

What you'll learn

1. Describing motion clearly

Scalars and vectors

Distance and displacement

Speed and velocity

Acceleration

2. Graphs as motion stories

Displacement, speed, velocity and acceleration graphs

3. Displacement–time graphs

What the shape tells you

4. Velocity–time graphs

Positive and negative areas

5. Estimating area under a non-linear velocity–time graph

6. Acceleration–time graphs

Review Kinematics by teaching Genie

Remember key concepts with flashcards

Kinematics

What you'll learn

1. Describing motion clearly

Scalars and vectors

Distance and displacement

Speed and velocity

Acceleration

2. Graphs as motion stories

Displacement, speed, velocity and acceleration graphs

3. Displacement–time graphs

What the shape tells you

4. Velocity–time graphs

Positive and negative areas

5. Estimating area under a non-linear velocity–time graph

6. Acceleration–time graphs

Review Kinematics by teaching Genie

Remember key concepts with flashcards

Kinematics Revision Guide

Kinematics

What you'll learn

1. Describing motion clearly

Scalars and vectors

Distance and displacement

Speed and velocity

Acceleration

2. Graphs as motion stories

Displacement, speed, velocity and acceleration graphs

3. Displacement–time graphs

What the shape tells you

4. Velocity–time graphs

Positive and negative areas

5. Estimating area under a non-linear velocity–time graph

6. Acceleration–time graphs

Review Kinematics by teaching Genie

Remember key concepts with flashcards

Kinematics Revision Guide