Velocity-time graphs

Velocity against time

Now plot the velocity, not the distance, against time. This velocity-time graph tells two things at once.

• A flat line: steady velocity, zero acceleration.
• A straight sloping line: steady acceleration.

The slope is the acceleration

A line sloping down means the object is slowing.

The area is the distance

The area between the line and the time axis is the distance travelled. For a flat line it is a rectangle, $v \times t$. For a straight sloping line from $u$ to $v$ it is a trapezium:

You should find that with $u = 0$ the area is a triangle, $\tfrac12 vt$, and that a steeper line (larger $a$) over the same time covers more distance.

Watch out. Read the axis label first. The same straight line means steady speed on a distance-time graph but steady acceleration on a velocity-time graph.

A worked example

A car's speed is read every $2\ \text{s}$: $0, 3, 6, 9, 9, 9\ \text{m/s}$. Find its acceleration at first and the distance in $10\ \text{s}$.

1. Slope. It gains $3\ \text{m/s}$ every $2\ \text{s}$: $a = 1.5\ \text{m/s}^2$ for the first $6\ \text{s}$.
2. Triangle. $\tfrac12 \times 6\ \text{s} \times 9\ \text{m/s} = 27\ \text{m}$.
3. Rectangle. $9\ \text{m/s} \times 4\ \text{s} = 36\ \text{m}$.
4. Total. $27\ \text{m} + 36\ \text{m} = 63\ \text{m}$.

Practise with new numbers

Try it in Python

The program adds up the area in thin strips.

Remember these

• On a velocity-time graph, the slope is the acceleration.
• The area under the line is the distance.
• Straight line from $u$ to $v$: $s = \dfrac{u + v}{2}\,t$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A velocity-time graph is a flat line at $8\ \text{m/s}$ for $5\ \text{s}$. How far does the object go?

2. A line goes from $0\ \text{m/s}$ to $12\ \text{m/s}$ in $4\ \text{s}$. What is the acceleration?

3. For the same line, how far does the object go?

4. What does a line sloping downward to the time axis show?