Velocity-Time Graphs

A velocity-time graph shows how an object's velocity changes over time.

Velocity-time graphs allow us to determine much more than how fast an object is moving. We can use them to identify:

• velocity at a particular time
• direction of motion
• constant velocity
• acceleration
• deceleration
• changes in direction

The slope (gradient) of a velocity-time graph represents the object's acceleration.

 

 

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Reading a Velocity-Time Graph

A velocity-time graph has two axes.

Horizontal Axis

The horizontal or x-axis represents:

Time (s)

Vertical Axis

The vertical or y-axis represents:

Velocity (m/s)

Unlike a speed-time graph, the vertical axis can contain both positive and negative values.

This is because velocity includes direction.

Velocity Includes Direction

Velocity describes both:

• how fast an object is moving
• the direction in which it is moving

We usually choose one direction to be positive.

For example:

Right = positive

Left = negative

An object travelling at:

+8 m/s

could therefore be moving right at 8 m/s.

An object travelling at:

−8 m/s

would be moving at the same speed but in the opposite direction.

The negative sign does not mean the object is moving slowly.

It tells us about its direction.

Positive and Negative Velocity

On a velocity-time graph:

Above the time axis → Positive velocity

Below the time axis → Negative velocity

On the time axis → Zero velocity

 

 

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For example:

Constant Velocity

A horizontal line on a velocity-time graph represents constant velocity.

For example, suppose a car travels at:

12 m/s for 5 seconds

Its velocity does not change.

Therefore:

Acceleration = 0 m/s²

The velocity-time graph would show a horizontal line at +12 m/s.

Constant velocity

An object travels at a constant velocity of 12 m/s.

A horizontal line therefore means:

Constant velocity → Zero acceleration

Acceleration and Slope

The slope of a velocity-time graph represents acceleration.

The equation is:

\( a = \frac{ \Delta v }{ \Delta t } = \frac{v_f - v_i}{t} \)

where:

• a = acceleration in m/s²
• v_f = final velocity in m/s
• v_i = initial velocity in m/s
• t = time in seconds

A steeper slope represents a larger magnitude of acceleration.

Calculating Acceleration from the Graph

Suppose an object's velocity increases from:

4 m/s to 16 m/s

during:

6 seconds

Step 1 – Find the change in velocity

Δv = 16 - 4 = 12 m/s

Step 2 – Divide by the time

\( a = \frac{12}{6} = 2 m/s^2 \)

The object accelerates at 2 m/s².

Positive Acceleration

Consider an object whose velocity changes as follows:

The line slopes upward.

Acceleration:

\( a = \frac{10 - 0}{5} = 2m/s^2 \)

The object has positive acceleration.

Negative Acceleration

Now suppose the velocity changes from:

20 m/s → 0 m/s

over 5 seconds.

\( a = \frac{0 - 20}{5} = -4m/s^2 \)

The negative acceleration means the velocity is changing in the negative direction.

In this particular example, the object has positive velocity but is slowing down, so we can also say it is decelerating.

 

 

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Deceleration

Deceleration means that an object's speed is decreasing.

For an object moving in the positive direction, deceleration appears as a line sloping downward toward zero.

For example:

15 m/s → 10 m/s → 5 m/s → 0 m/s

The object becomes progressively slower until it stops.

However, we need to be careful:

Negative acceleration does not always mean deceleration.

An object moving in the negative direction can have negative acceleration and actually speed up.

Acceleration vs Deceleration

The easiest way to determine whether an object is speeding up or slowing down is to look at the magnitude of its velocity.

Moving Away from Zero

Speed is increasing.

Object is speeding up.

Moving Toward Zero

Speed is decreasing.

Object is slowing down.

For example:

−2 m/s → −4 m/s → −6 m/s

The values are becoming more negative, but the object's speed is increasing:

2 m/s → 4 m/s → 6 m/s

So the object is speeding up in the negative direction.

Crossing the Time Axis

One of the most important features of a velocity-time graph occurs when the line crosses:

v = 0

At that instant, the object has zero velocity.

If the velocity then changes sign, the object has changed direction.

For example:

+6 m/s → +3 m/s → 0 m/s → −3 m/s → −6 m/s

The object:

1. moves in the positive direction
2. slows down
3. momentarily stops
4. reverses direction
5. speeds up in the negative direction

 

 

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This is a major difference between speed-time graphs and velocity-time graphs.

A speed-time graph cannot have negative speed.

A velocity-time graph can have negative velocity.

Interpreting Different Sections

A velocity-time graph may contain several different sections.

Imagine the following journey:

Section A

Velocity increases from:

0 → +10 m/s

The object is accelerating in the positive direction.

Section B

Velocity remains:

+10 m/s

The object travels at constant positive velocity.

Section C

Velocity decreases:

+10 → 0 m/s

The object decelerates to a stop.

Section D

Velocity becomes:

0 → −5 m/s

The object changes direction and speeds up in the negative direction.

Section E

Velocity remains:

−5 m/s

The object moves at constant velocity in the negative direction.

A Complete Journey

Consider this example:

We can analyse each section.

0–2 seconds

Velocity:

0 → +6 m/s

The object accelerates.

\( a = \frac{6 - 0}{2} = 3m/s^2 \)

2–4 seconds

Velocity remains:

+6 m/s

The object moves at constant positive velocity.

a = 0 m/s²

4–6 seconds

Velocity:

+6 → 0 m/s

\( a = \frac{0 - 6}{2} = -3m/s^2 \)

The object slows to a stop.

6–8 seconds

Velocity:

0 → −4 m/s

\( a = \frac{-4 - 0}{2} = -2m/s^2 \)

The object accelerates in the negative direction.

8–10 seconds

Velocity remains:

−4 m/s

The object moves at a constant velocity in the negative direction.

a = 0 m/s²

Understanding the Slope

The slope tells us how rapidly velocity is changing.

Remember:

Slope = acceleration

This is one of the most important relationships when interpreting a velocity-time graph.

Constructing a Velocity-Time Graph

Suppose we are given the following description:

A cyclist starts from rest and accelerates uniformly to 8 m/s in 4 seconds. The cyclist travels at 8 m/s for another 3 seconds before slowing uniformly to rest over 2 seconds.

First, create a table.

Then:

1. put time on the x-axis
2. put velocity on the y-axis
3. choose an appropriate scale
4. plot the points
5. connect the points with straight lines

Analysing the Cyclist's Graph

0–4 seconds

The cyclist accelerates:

\( a = \frac{8 - 0}{4} = 2m/s^2 \)

4–7 seconds

Velocity remains constant at: 8m/s

Therefore:

a = 0 m/s²

7–9 seconds

The cyclist slows:

\( a = \frac{0 - 8}{2} = -4m/s^2 \)

Notice that the final section is steeper than the first.

Therefore, the magnitude of the cyclist's deceleration is greater than the magnitude of the initial acceleration.

Velocity-Time Graphs and Real Motion

Velocity-time graphs can describe many real situations.

For example, a car approaching traffic lights might:

Accelerate → Constant velocity → Decelerate → Stop

An elevator might:

Accelerate upward → Constant upward velocity → Decelerate → Stop

Then later:

Accelerate downward → Constant negative velocity → Decelerate → Stop

 

 

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A Good Strategy for Reading Any Velocity-Time Graph

When you see a velocity-time graph, work through it systematically.

First: Look at whether the graph is above or below zero.

This tells you the direction of motion.

Second: Look at the slope.

This tells you the acceleration.

Third: Look for horizontal sections.

These represent constant velocity.

Fourth: Look for places where the graph reaches or crosses zero.

These may represent the object stopping or changing direction.

Finally: Calculate slopes when numerical acceleration values are required.

Common Misconception

A downward-sloping line does not always mean the object is slowing down.

Consider:

0 → −5 → −10 m/s

The graph slopes downward, so acceleration is negative.

But the object's speed changes:

0 → 5 → 10 m/s

The object is actually speeding up in the negative direction.

Therefore:

Negative acceleration ≠ always slowing down

Instead, compare the direction of velocity and acceleration.

Did You Know?

Velocity-time graphs can also tell us an object's displacement.

The displacement during a time interval is equal to the signed area between the graph and the time axis.

Areas above the axis represent displacement in the positive direction.

Areas below the axis represent displacement in the negative direction.

This makes velocity-time graphs especially powerful: the slope tells us acceleration, while the area tells us displacement.

Key Terms

Velocity-time graph – A graph showing how velocity changes with time.

Velocity – Speed in a specified direction.

Acceleration – Rate of change of velocity.

Deceleration – A decrease in speed.

Slope – The steepness of a graph; on a velocity-time graph it represents acceleration.

Constant velocity – Motion with unchanged velocity.

Positive velocity – Motion in the chosen positive direction.

Negative velocity – Motion in the direction opposite to the chosen positive direction.

Zero velocity – The object is stationary at that instant.

Key Takeaways

• A velocity-time graph shows how velocity changes with time.
• Time is plotted on the x-axis and velocity on the y-axis.
• The slope of a velocity-time graph represents acceleration.
• Acceleration can be calculated using \( a = \frac{ \Delta v }{ \Delta t } = \frac{v_f - v_i}{t} \).
• A horizontal line represents constant velocity and zero acceleration.
• An upward slope represents positive acceleration.
• A downward slope represents negative acceleration.
• Velocity above the time axis is positive.
• Velocity below the time axis is negative.
• Negative velocity means movement in the opposite direction, not negative speed.
• If the graph crosses the time axis, the object may be changing direction.
• Deceleration means speed is decreasing, so negative acceleration does not always mean deceleration.
• Velocity-time graphs can be constructed from motion descriptions or numerical data.
• The slope gives acceleration, while the signed area under the graph gives displacement.