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Acceleration

Change and motion · level 5, lesson 4 of 6

A question to start

Car adverts love one number: "0 to 100 km/h in 8 seconds". A car that does it in 4 seconds is called fast; one that takes 15 seconds is called slow. But wait: all of them reach the same speed, 100 km/h. So the advert is not about speed at all.

What is it about? It is about how quickly the speed itself changes. That is a rate of change of a rate of change, and it has its own name: acceleration. How much faster does the 8-second car get in each second? And what strange unit, "metres per second squared", hides inside that answer?

The idea

Speed can change too

So far a rate has told us how fast something changes: metres per second, dollars per item. Speed is itself a quantity that can change. A bike starting off goes from 0 m/s to 3 m/s to 6 m/s; a car braking for a red light goes from 15 m/s down to 0.

Acceleration is the rate of change of speed:

\[\text{acceleration} = \frac{\text{change in speed}}{\text{change in time}}.\]

(To be careful, scientists use the change in velocity, the speed with a sign for direction, from lesson 2. In this lesson everything moves in one direction, so speed and velocity are the same number.)

The units: metres per second, per second

Suppose a cyclist's speed goes from 4 m/s to 10 m/s in 3 seconds. The change in speed is \(10 - 4 = 6\) m/s, so

\[\text{acceleration} = \frac{6 \text{ m/s}}{3 \text{ s}} = 2 \text{ m/s per second}.\]

"2 metres per second, per second" means: each second, the speed goes up by 2 m/s.

Time (s) 0 1 2 3
Speed (m/s) 4 6 8 10

Saying "metres per second per second" is long, so it is written \(\text{m/s}^2\), read "metres per second squared". The "squared" comes only from dividing by seconds twice:

\[\frac{\text{m/s}}{\text{s}} = \frac{\text{m}}{\text{s} \times \text{s}} = \frac{\text{m}}{\text{s}^2}.\]

There is no "square second" you can picture. Always translate it back in your head: so many m/s gained each second.

The car advert

The car goes from 0 to 100 km/h in 8 s. First change the speed to metres per second: 100 km/h is 100 000 m in 3600 s, which is

\[100\,000 \div 3600 \approx 27.8 \text{ m/s}.\]

Then

\[\text{acceleration} \approx \frac{27.8 \text{ m/s}}{8 \text{ s}} \approx 3.47 \text{ m/s}^2.\]

Each second, the car gains about 3.5 m/s, or \(100 \div 8 = 12.5\) km/h. Both answers describe the same thing: one is in km/h per second, the other in m/s per second.

On a speed-time graph, acceleration is the slope

Put time across and speed up. Then the rise is the change in speed and the run is the change in time, so the slope is change in speed over change in time: the acceleration. It is the same move as in lesson 1 (slope of a distance-time graph is speed), one level up.

Here is a short car trip.

PQR 010202530 time (seconds) 0510152025 m/s
Part Change in speed Time Acceleration Story
P \(20 - 0 = 20\) m/s 10 s \(+2\) m/s² speeding up
Q \(20 - 20 = 0\) m/s 15 s \(0\) m/s² steady speed
R \(0 - 20 = -20\) m/s 5 s \(-4\) m/s² braking

Three lessons in one table:

  • Flat on a speed-time graph means steady speed, not standing still. (On a distance-time graph, flat meant standing still. Always read the label on the up axis.)
  • A negative acceleration means the speed is going down. People often call it deceleration: the car decelerates at 4 m/s².
  • A steeper piece means a bigger acceleration. The brakes (R) change the speed twice as fast as the engine (P).

Constant acceleration on a distance-time graph

What does steady acceleration look like on a distance-time graph? Take a car starting from rest with acceleration 2 m/s². Its speed after \(t\) seconds is \(2t\) m/s, and (you will see why in lesson 6) the distance it has covered is \(t^2\) metres.

Time (s) 0 1 2 3 4 5
Distance (m) 0 1 4 9 16 25
Distance in that second (m) 1 3 5 7 9

The distance covered in each second grows: 1, 3, 5, 7, 9. Those numbers go up by exactly 2 each time, the acceleration. Steady acceleration makes a straight line on a speed-time graph but a curve that keeps getting steeper on a distance-time graph. The falling stone from lesson 3 is exactly like this, with the numbers 4.9, 14.7, 24.5, 34.3 going up by 9.8 each second.

Gravity is an acceleration

That 9.8 is no accident. Near the Earth's surface, anything falling freely (with air resistance too small to matter) gains about 9.8 m/s of speed every second. This is called the acceleration due to gravity, often written \(g\):

\[g \approx 9.8 \text{ m/s}^2 \text{ on Earth}.\]

A heavy stone and a light stone dropped together gain speed at the same rate (if the air does not interfere). On Mars, which is smaller and pulls less strongly, \(g\) is about 3.71 m/s². The next lesson is all about falling.

Worked examples

1. A sprinter. A sprinter goes from 0 to 9 m/s in the first 3 s of a race. Find her average acceleration.

\(\dfrac{9 - 0}{3} = 3\) m/s². (Real sprinters speed up fastest at the very start, so 3 m/s² is an average over those 3 seconds.)

2. Braking. A bus moving at 15 m/s brakes steadily and stops in 6 s. Find the acceleration.

\(\dfrac{0 - 15}{6} = -2.5\) m/s². The bus loses 2.5 m/s of speed every second.

3. How long to reach a speed? A train accelerates at 0.5 m/s² from rest. How long until it reaches 30 m/s (108 km/h)?

Each second it gains 0.5 m/s, so it needs \(30 \div 0.5 = 60\) seconds, one minute.

4. Speed after a time. A cyclist going 3 m/s accelerates at 1.5 m/s² for 4 s. What is her new speed?

Gain \(= 1.5 \times 4 = 6\) m/s. New speed \(= 3 + 6 = 9\) m/s.

5. Thrown upwards. A ball is thrown straight up at 20 m/s. Gravity takes away 9.8 m/s of upward speed every second. When does it stop rising?

After 1 s its upward speed is \(20 - 9.8 = 10.2\) m/s; after 2 s, \(20 - 19.6 = 0.4\) m/s. It stops when \(20 - 9.8t = 0\), so \(t = 20 \div 9.8 \approx 2.04\) s. At that instant its speed is zero, but its acceleration is still \(-9.8\) m/s²: gravity never switches off, which is why it starts falling back.

Try it

  1. A scooter goes from 2 m/s to 8 m/s in 4 s. Find its acceleration.
  2. A car moving at 30 m/s brakes to 10 m/s in 5 s. Find its acceleration, with the right sign.
  3. A motorbike goes from 0 to 100 km/h in 4 s. Find its acceleration in km/h per second and in m/s² (to 2 decimal places).
  4. Starting from rest, a lift accelerates at 1.2 m/s². What is its speed after 2.5 s?
  5. On a speed-time graph, a line goes from (0 s, 12 m/s) to (8 s, 12 m/s). What is the acceleration? What is the car doing?
  6. A plane needs 70 m/s to take off and accelerates at 2.5 m/s² from rest. How long does its run take?
  7. A car has speed \(3t\) m/s after \(t\) seconds, starting from rest; its distance is \(1.5t^2\) m. Make a table of the distance at \(t\) = 0, 1, 2, 3, 4, and of the distance covered in each second. What do you notice?
  8. A stone is dropped on Mars, where it gains 3.71 m/s every second. How fast is it falling after 3 s? How long would it take to reach the speed a stone on Earth reaches in 1 s?
Answers
  1. \((8 - 2) \div 4 = 1.5\) m/s².
  2. \((10 - 30) \div 5 = -20 \div 5 = -4\) m/s² (slowing down).
  3. \(100 \div 4 = 25\) km/h per second. In m/s: \(100 \div 3.6 \approx 27.78\) m/s, so \(27.78 \div 4 \approx 6.94\) m/s².
  4. \(1.2 \times 2.5 = 3\) m/s.
  5. Change in speed \(= 0\), so acceleration \(= 0\) m/s². The car is moving at a steady 12 m/s (not stopped).
  6. \(70 \div 2.5 = 28\) seconds.
  7. Distances: 0, 1.5, 6, 13.5, 24. In each second: 1.5, 4.5, 7.5, 10.5. These go up by 3 each time, which is the acceleration (3 m/s²).
  8. \(3.71 \times 3 = 11.13\) m/s. An Earth stone reaches 9.8 m/s after 1 s; on Mars that takes \(9.8 \div 3.71 \approx 2.64\) s.

Common mistake

Thinking zero speed means zero acceleration. At the top of its flight a ball thrown up is still for an instant, but its speed is changing (from going up to going down), so its acceleration is \(-9.8\) m/s², not 0.

Thinking a flat speed-time graph means stopped. Flat means the speed is not changing. The object is moving at a steady speed. Only a graph that sits on the time axis (speed 0) means stopped.

Mixing units. In "0 to 100 km/h in 8 s", dividing 100 by 8 gives 12.5 km/h per second, not m/s². Turn km/h into m/s first (divide by 3.6) if you want m/s².

Where it leads