2.1.7Motion and Forces

Velocity-Time Graphs

5 min read 8 parts Lesson 8 of 24 1 quiz

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Section 1 of 8What the graph shows

A velocity-time graph plots velocity (m/s) on the vertical axis against time (s) on the horizontal axis.

What the graph shows

It tells us how the velocity of an object changes during a journey.

It does not show where the object is, which is the job of a distance-time graph (1.1.1).

You can find the acceleration from the gradient, and the distance travelled from the area under the line (1.1.6).

Section 2 of 8Reading the shape

1Horizontal line

A horizontal line at a value above zero: the velocity is constant, so the acceleration is zero (the object moves at a steady speed).

2On the time axis

A horizontal line on the time axis (velocity 0): the object is stationary.

3Straight line sloping upwards

A straight line sloping upwards: constant (uniform) acceleration. A steeper line is a larger acceleration.

4Straight line sloping downwards

A straight line sloping downwards: constant deceleration (negative acceleration).

5Curve

A curve getting steeper: the acceleration is increasing; a curve getting flatter: the acceleration is decreasing.

For a falling object with air resistance, the curve flattens and becomes horizontal at the terminal velocity.

6Below the time axis

A line that goes below the time axis means that the velocity is negative, that is, in the opposite direction.

Be carefula velocity-time graph that is horizontal does not mean the object is stationary, unless the line is at zero.

Section 3 of 8Gradient equals acceleration

The gradient of a velocity-time graph is the change in velocity divided by the change in time, which is exactly the definition of acceleration:

acceleration = gradient = Δv ÷ Δt

For a straight-line section, draw a right-angled triangle against the line, read off the vertical change (Δv, in m/s) and the horizontal change (Δt, in s) and divide. The unit is m/s². For a curve, the acceleration at a moment is the gradient of the tangent at that point.

A downwards-sloping line gives a negative gradient and therefore a negative acceleration (deceleration).

Section 4 of 8Describing a whole journey

Most journeys are split into sections.

For example, a car pulling away from traffic lights shows (a) an upwards sloping line as it accelerates from rest, (b) a horizontal line as it cruises at a steady speed,

and (c) a downwards sloping line as it brakes to a stop.

To describe these in words, give for each section the velocities at the start and end, whether the acceleration is positive, zero or negative,

and how long it lasted.

You may be asked to compare sections: the steeper line has the larger magnitude of acceleration.

Remember also that distance travelled is the area under the graph; the next lesson shows you how to work this out.

Section 5 of 8Drawing a velocity-time graph

To draw a graph from a description, first decide on scales for the axes, so that the data fills most of the grid.

Plot each given point with a small cross, and join the points in straight lines if the acceleration is described as uniform.

For example, a car starts from rest and accelerates uniformly to 20 m/s in 10 s, drives at 20 m/s for 30 s, and brakes uniformly to rest in 5 s.

The points are (0, 0), (10, 20), (40, 20) and (45, 0).

The graph is a trapezium.

The accelerations are 2.0 m/s² for the first stage, zero for the second and −4.0 m/s² for the third.

Label the axes with quantities and units, for example 'Velocity / m/s' and 'Time / s'.

Drawing a velocity-time graph

Compare the distance-time graph for the same journey: it would curve upwards in the first section, be a straight line in the second,

and curve to flat in the last.
Being able to convert between the two types of graph is a good test of understanding.
Look at 1.1.1 and 1.1.6 for the matching ideas.

As a check on any graph answer, confirm that your acceleration has the right sign and a sensible size, and that the unit is m/s² and not m/s.

If two graphs are drawn for different objects on the same axes, the one with the steeper line has the larger acceleration,

and where the lines cross the two objects have the same velocity at that instant, though not necessarily the same position.

Section 6 of 8Worked examples

A velocity-time graph is a straight line from (0 s, 0 m/s) to (6.0 s, 15 m/s). Find the acceleration.
A cyclist is moving at 10 m/s. The graph slopes down in a straight line to 0 m/s after 8.0 s. Find the acceleration.
A graph has three sections: 0–4 s a line rising from 0 to 8 m/s; 4–10 s horizontal at 8 m/s; 10–12 s falling from 8 m/s to 0. Find the acceleration in each section.

Section 7 of 8Key terms

Term	Meaning
Velocity-time graph	A graph showing how velocity varies with time.
Gradient	Δv ÷ Δt = acceleration.
Uniform acceleration	Constant acceleration; a straight sloping line.
Terminal velocity	Constant velocity when forces balance; a horizontal line at the end of a curve.
Deceleration	Negative acceleration; a downward-sloping line.

Section 8 of 8Exam technique

Reading a horizontal line as 'stopped'. It is constant velocity unless it is at zero.

Reading the graph like a distance-time graph.

Using the vertical value as the acceleration instead of the gradient.

Forgetting to give negative acceleration for a downward slope.

Making the triangle too small when finding the gradient, which reduces accuracy.

Physics 0625 & 9702 SUVAT Mastery Graph Interpretation

Constant Acceleration Equations (SUVAT) & Slopes

v = u + at | s = ut + ½at² | v² = u² + 2as | s = ½(u + v)t

Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
Free Fall in vacuum: a = g ≈ 9.81 m/s² downwards (air resistance negligible).