Acceleration and velocity–time graphs

What is acceleration?

Speed tells you how fast something is going. Acceleration tells you how quickly its velocity is changing. Remember that velocity is speed with a direction.

Why $\mathrm{m/s^2}$? If a car gains $3\,\mathrm{m/s}$ of velocity every second, its acceleration is "3 metres per second, per second", or $3\,\mathrm{m/s^2}$.

Answer to "Your turn": $a = \dfrac{6-0}{2} = 3\,\mathrm{m/s^2}$.

If the velocity is not changing, the acceleration is zero. A car at a steady $20\,\mathrm{m/s}$ has $a = 0$, even though it is moving fast.

Velocity–time graphs

A velocity–time graph plots time on the horizontal axis and velocity on the vertical axis. Label the axes with the quantity and unit: Time (s) and Velocity (m/s).

Read the graph like this:

Slope = acceleration. The slope is the change in velocity divided by the change in time, which is exactly the formula $a = \dfrac{v-u}{t}$. A steeper line means a bigger acceleration.

Answer to "Your turn": the line is horizontal, so $a = 0$.

A downward slope gives a negative acceleration, sometimes called deceleration. It means the velocity is decreasing. It does not mean the object is going backwards.

Distance from the area under the graph

For constant velocity, distance = speed × time. On a velocity–time graph, velocity is the height and time is the width, so height × width is the area. The area under the graph is the distance travelled. When the velocity changes, split the graph into rectangles and triangles and add their areas.

• Rectangle: area = base × height.

• Triangle: area = $\tfrac{1}{2}$ × base × height.

• For a graph with several stages, split it into triangles and rectangles and add the areas.

Answer to "Your turn": $360 + 1440 = 1800\,\mathrm{m}$.

For average speed, use total distance ÷ total time. For the train, $1800 \div 90 = 20\,\mathrm{m/s}$. It is not the average of the speeds $0$ and $24\,\mathrm{m/s}$, which would be $12\,\mathrm{m/s}$.

Putting it all together

A full journey often has three stages: speeding up, constant velocity, slowing down. For each stage, find the acceleration from the slope; for the distance, add up the areas.

Answer to "Your turn": $600 \div 50 = 12\,\mathrm{m/s}$.

To compare two objects, work out each acceleration with $a = \dfrac{v-u}{t}$ and compare the numbers. Car A: $30 \div 10 = 3\,\mathrm{m/s^2}$. Car B: $24 \div 6 = 4\,\mathrm{m/s^2}$. B has the greater acceleration, even though A reaches the higher final velocity.

Back to the Luas: it speeds up (upward slope), cruises (flat line) and brakes (downward slope), and the area under that graph is how far you travelled.

Summary