Acceleration and velocity-time graphs

Faster, slower, turning round

A car at a signal moves off: its velocity grows. A cyclist brakes for a puddle: her velocity shrinks. A ball rolled up a slope stops and rolls back: its velocity goes through zero and becomes negative. In all three, the velocity changes. How fast it changes is the acceleration.

Acceleration

Here $u$ is the velocity at the start, $v$ the velocity after a time $t$. The unit is $\text{m/s}$ per second, written $\text{m/s}^2$.

A worked example. An e-rickshaw speeds up steadily from $4\ \text{m/s}$ to $10\ \text{m/s}$ in $3\ \text{s}$.

1. Acceleration. $a = \dfrac{10\ \text{m/s} - 4\ \text{m/s}}{3\ \text{s}} = 2\ \text{m/s}^2$: it gains $2\ \text{m/s}$ every second.
2. Distance. With a steady acceleration, the average velocity is $\dfrac{4\ \text{m/s} + 10\ \text{m/s}}{2} = 7\ \text{m/s}$, so it covers $7\ \text{m/s} \times 3\ \text{s} = 21\ \text{m}$.

The sign of the acceleration

Along a line, acceleration has a sign, like velocity. The rule that students most often get wrong:

• If $v$ and $a$ have the same sign, the body speeds up.
• If they have opposite signs, the body slows down.

A negative acceleration does not always mean slowing down. A car reversing (negative velocity) and getting faster has a negative acceleration too.

Think first. A ball has $v = +5\ \text{m/s}$ and $a = -2\ \text{m/s}^2$. What happens? It slows down, stops after $2.5\ \text{s}$, and then moves the other way, faster and faster.

The velocity-time graph

Plot velocity up the side and time along the bottom. Two facts make this graph the most useful one in mechanics.

• The slope is the acceleration. A straight line means a steady acceleration. A flat line means zero acceleration: a steady velocity.
• The area under the line is the displacement. Each thin strip of the area is velocity $\times$ a short time, which is a small displacement. Adding the strips gives the whole displacement. Area below the time axis counts as negative: the body moved backwards then.

Try both ideas here.

Watch out. The height of a velocity-time graph is a velocity, not a position. A line going down does not mean the body is coming back: it means the body is slowing down (while the line is above the axis). The body comes back only when the line goes below the axis.

Practise with new numbers

Try it in Python

A bus's speedometer is read every $5\ \text{s}$. The program adds up the area under the velocity-time graph, strip by strip, and finds the slope of each stretch.

Remember these

• $a = \dfrac{v - u}{t}$, in $\text{m/s}^2$.
• Same signs: speeding up. Opposite signs: slowing down.
• Velocity-time graph: slope = acceleration, area = displacement (negative below the axis).

Check yourself

1. A motorbike goes from $6\ \text{m/s}$ to $18\ \text{m/s}$ in $4\ \text{s}$. Find its acceleration.

2. A trolley has $v = +5\ \text{m/s}$ and $a = -2\ \text{m/s}^2$. Is it speeding up or slowing down? What happens after $2.5\ \text{s}$?

3. A velocity-time graph is a flat line at $7\ \text{m/s}$ for $9\ \text{s}$. What are the acceleration and the distance?

4. What does an area below the time axis on a velocity-time graph mean?