Describing motion along a line

What you'll learn

How to describe motion using distance, displacement, speed, velocity and acceleration.
How to calculate average speed and acceleration using GCSE equations.
How to interpret distance-time graphs and velocity-time graphs.
How to use graph gradients and areas to find useful motion quantities.

The starting idea: motion in one dimension

In this topic, we simplify motion by looking at objects moving along a line. That might mean a car moving along a straight road, a runner on a straight track, or a trolley moving along a bench.

Definition

Motion along a line

Motion along a line means the object’s position changes along one straight path, so directions can be described simply as forwards/backwards, left/right, or positive/negative.

To describe motion clearly, you need to know both how far something moves and sometimes which direction it moves in.

Scalars and vectors

Some quantities only need a size. Others need a size and a direction.

Definition

Scalars and vectors

A scalar quantity has size only, such as distance or speed.
A vector quantity has size and direction, such as displacement, velocity or acceleration.

For motion along a line, direction is often shown using signs. For example, if forwards is chosen as positive, backwards can be negative.

Distance and displacement

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

Displacement is how far an object is from its starting point in a particular direction. It is a vector, so direction matters.

Example

Comparing distance and displacement

A student walks 6 m forwards, then 2 m backwards.

The total path length is found by adding both parts of the journey: 6 m plus 2 m, so the distance travelled is 8 m.
The final position compared with the start is 6 m forwards minus 2 m backwards, giving 4 m forwards.
The displacement is therefore 4 m forwards, or $m+4\ \text{m}$ if forwards is the positive direction.

Key Idea

Distance is not the same as displacement

Distance counts the whole route travelled. Displacement only compares the final position with the starting position, including direction.

Speed: how quickly distance is covered

Speed tells you how fast something is moving. More precisely, it is the distance travelled per unit time.

The GCSE equation is:

\frac{s}{t}

where:

$v$ is speed in metres per second, m/s
$s$ is distance in metres, m
$t$ is time in seconds, s

This gives the average speed over the journey. The object may have sped up or slowed down during the journey, but the calculation spreads the motion evenly over the whole time.

Example

Calculating average speed

A cyclist travels 300 m in 50 s. Calculate the average speed.

Choose the speed equation because you are given distance and time: $\frac{s}{t}$ .
Substitute the values with units: $\frac{300\ \text{m}}{50\ \text{s}}$ .
Calculate the result: $6\ \text{m/s}$ , so the cyclist’s average speed is 6 m/s.

Common Mistake

Mixing up minutes and seconds

If the time is given in minutes, convert it to seconds before using $\frac{s}{t}$ . For example, 2 minutes is 120 s.

Typical speeds

You are not expected to memorise every possible speed, but you should have a feel for sensible values. A walking speed is about 1.5 m/s, running is a few metres per second, and the speed of sound in air is about 330 m/s.

Tip

Sanity check your answer

If you calculate a walking speed of 80 m/s, something has probably gone wrong with units or arithmetic.

Velocity: speed in a stated direction

Velocity is speed in a particular direction. It is a vector.

For motion along a line, velocity can be positive or negative depending on the direction chosen. If forwards is positive, then:

$+10\ \text{m/s}$ means 10 m/s forwards
$-10\ \text{m/s}$ means 10 m/s backwards

A change in velocity can happen because the object changes speed, changes direction, or both.

Acceleration: changing velocity

Acceleration is the rate of change of velocity. In simpler words, it tells you how quickly the velocity changes.

The GCSE equation is:

\frac{\Delta v}{t}

You may also see this written as:

\frac{v - u}{t}

where $u$ is the initial velocity and $v$ is the final velocity.

Acceleration is measured in metres per second squared, m/s². This unit means “metres per second, per second”: the velocity changes by that many m/s every second.

Example

Calculating acceleration

A car’s velocity increases from 4 m/s to 20 m/s in 8 s. Calculate its acceleration.

Find the change in velocity: $m/s\Delta v = 20\ \text{m/s} - 4\ \text{m/s} = 16\ \text{m/s}$ .
Substitute into the acceleration equation: $\frac{16\ \text{m/s}}{8\ \text{s}}$ .
Calculate the acceleration: $2\ \text{m/s}^2$ , so the car’s velocity increases by 2 m/s every second.

Deceleration

Deceleration usually means slowing down. In physics, this is often acceleration in the opposite direction to the motion.

If forwards is positive, a car moving forwards and slowing down has a negative acceleration.

Common Mistake

Acceleration does not always mean speeding up

Acceleration means changing velocity. An object can accelerate by speeding up, slowing down, or changing direction.

Distance-time graphs

A distance-time graph shows how the distance travelled changes with time.

Time goes on the horizontal axis. Distance goes on the vertical axis.

Distance-time graph showing constant speed, stationary and faster sections

The key rule is:

graph\text{speed} = \text{gradient of a distance-time graph}

On a distance-time graph:

A straight sloping line means constant speed.
A horizontal line means the object is stationary.
A steeper line means a greater speed.
A curve means the speed is changing.

Example

Finding speed from a distance-time graph

On a distance-time graph, an object moves from 20 m at 5 s to 80 m at 17 s. Calculate its speed during this straight-line section.

Find the change in distance: $m80\ \text{m} - 20\ \text{m} = 60\ \text{m}$ .
Find the change in time: $s17\ \text{s} - 5\ \text{s} = 12\ \text{s}$ .
Use gradient: $m/s\text{speed} = \frac{60\ \text{m}}{12\ \text{s}} = 5\ \text{m/s}$ .

Tip

Use a large gradient triangle

When finding a gradient from a graph, use two clear points far apart on the line. This reduces the effect of small reading errors.

Common Mistake

Reading the height as the speed

On a distance-time graph, the vertical value is distance, not speed. Speed comes from the gradient.

Velocity-time graphs

A velocity-time graph shows how velocity changes with time.

Time goes on the horizontal axis. Velocity goes on the vertical axis.

Velocity-time graph showing acceleration, constant velocity and area under the graph

There are two very important rules:

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

graph\text{distance travelled} = \text{area under a velocity-time graph}

On a velocity-time graph:

A horizontal line means constant velocity, not necessarily stationary.
A line sloping upwards means positive acceleration.
A line sloping downwards means negative acceleration.
A line on the time axis, where velocity is zero, means the object is stationary.

Example

Using gradient on a velocity-time graph

A train’s velocity increases from 0 m/s to 30 m/s in 15 s. Calculate its acceleration.

Find the change in velocity: $m/s30\ \text{m/s} - 0\ \text{m/s} = 30\ \text{m/s}$ .
Use the gradient rule for a velocity-time graph: $\frac{\Delta v}{t}$ .
Substitute and calculate: $\frac{30\ \text{m/s}}{15\ \text{s}} = 2\ \text{m/s}^2$ .

Example

Finding distance from a velocity-time graph

A car accelerates from 0 m/s to 12 m/s in 4 s, then travels at 12 m/s for 6 s. Find the distance travelled.

Split the area under the graph into a triangle for the acceleration section and a rectangle for the constant velocity section.
Calculate the triangle area: $m\frac{1}{2} \times 4\ \text{s} \times 12\ \text{m/s} = 24\ \text{m}$ .
Calculate the rectangle area: $m6\ \text{s} \times 12\ \text{m/s} = 72\ \text{m}$ , so the total distance is $m24\ \text{m} + 72\ \text{m} = 96\ \text{m}$ .

Common Mistake

Below the time axis

If velocity is below the time axis, the object is moving in the opposite direction. Areas below the axis count as negative displacement, but exam questions often ask for total distance travelled, so read the wording carefully.

Choosing the right graph rule

A quick way to remember the graph skills is:

Key Idea

Graph rules for motion

On a distance-time graph, gradient gives speed.
On a velocity-time graph, gradient gives acceleration.
On a velocity-time graph, area under the graph gives distance travelled or displacement.

The word “gradient” means “slope”. For a straight line, calculate it using:

value\text{gradient} = \frac{\text{change in vertical value}}{\text{change in horizontal value}}

The units help you check the rule:

distance divided by time gives m/s, so a distance-time gradient is speed
velocity divided by time gives m/s², so a velocity-time gradient is acceleration
velocity multiplied by time gives m, so velocity-time area is distance or displacement

Exam technique

In the exam

Check the graph axes first: distance-time and velocity-time graphs use different rules.
For gradients, use two points on the line and calculate change in vertical value divided by change in horizontal value.
For areas under velocity-time graphs, split the shape into rectangles and triangles, then include the correct units.

Self review

Check yourself

What is the difference between distance and displacement?
On a distance-time graph, what does a horizontal line mean?
How would you find the distance travelled from a velocity-time graph?

Describing motion along a line

What you'll learn

How to describe motion using distance, displacement, speed, velocity and acceleration.
How to calculate average speed and acceleration using GCSE equations.
How to interpret distance-time graphs and velocity-time graphs.
How to use graph gradients and areas to find useful motion quantities.

The starting idea: motion in one dimension

In this topic, we simplify motion by looking at objects moving along a line. That might mean a car moving along a straight road, a runner on a straight track, or a trolley moving along a bench.

Definition

Motion along a line

Motion along a line means the object’s position changes along one straight path, so directions can be described simply as forwards/backwards, left/right, or positive/negative.

To describe motion clearly, you need to know both how far something moves and sometimes which direction it moves in.