S4.8.2 Chapter 2

Velocity and acceleration

A bus leaves the stop and picks up speed. How do we say how quickly it speeds up?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A By how much its velocity changes each second.
You picked A. Read on. At the end, the chapter comes back to your guess.
B By the distance it covers each second.
You picked B. Read on. At the end, the chapter comes back to your guess.
C By the top speed it reaches.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate acceleration from change in velocity and time
Find acceleration and distance from a velocity and time graph

In the last chapter you met vectors. Velocity is speed in a stated direction, so it is a vector. Its unit is the metre per second, m/s.

A car at 20 m/s east and a car at 20 m/s west have the same speed. They have different velocities. If east is positive, the second car has a velocity of −20 m/s.

Something accelerates when its velocity changes. It may speed up or slow down. It also accelerates when it turns, because a new direction is a new velocity. In this chapter we stay on a straight line.

The acceleration is the change in velocity divided by the time taken. We write u for the first velocity and v for the last one. Then a = (v − u) ÷ t, where t is the time in seconds.

A bus that gains 2 m/s every second has an acceleration of 2 m/s per second. We write this as 2 m/s², and say "two metres per second squared".

Draw it first

A bus gains 2 m/s every second

1 At 0 s the bus is at rest. After 1 s its velocity is 2 m/s. Mark both dots.
2 After 2 s it is 4 m/s, after 3 s it is 6 m/s and after 4 s it is 8 m/s. Mark each dot.
3 The dots lie on a straight line. Equal gains in velocity each second always make a straight line.

The steepness of the line is its gradient: the rise divided by the run. On this graph the rise is a change in velocity and the run is a time. So the gradient is (change in velocity) ÷ (time), and that is the acceleration.

Here the gradient is 8 m/s ÷ 4 s = 2 m/s². The unit follows from the axes: metres per second, divided by seconds, gives metres per second squared.

See it three ways

As a picture

In words

The velocity rises by the same amount each second, so the line is straight. The gradient of the line is the acceleration.

In symbols

a = (v − u) ÷ t = (8 − 0) ÷ 4 = 2 m/s²

Worked example

A car starts from rest and reaches 20 m/s in 8 s. Find its acceleration.

Write the rule: a = (v − u) ÷ t. Here u = 0 m/s, v = 20 m/s and t = 8 s.
The change in velocity is 20 m/s − 0 m/s = 20 m/s.
a = 20 m/s ÷ 8 s = 2.5 m/s².

Answer: 2.5 m/s²

The line rises 20 m/s in 8 s. Its gradient is the acceleration, 2.5 m/s².

When something slows down in the positive direction, its velocity falls, so the change in velocity is negative. The acceleration is then negative. We often call this deceleration. It means the acceleration points the opposite way to the positive direction.

Worked example

A cyclist slows from 12 m/s to 4 m/s in 4 s. Take her direction of travel as positive. Find her acceleration.

Here u = 12 m/s, v = 4 m/s and t = 4 s.
The change in velocity is 4 m/s − 12 m/s = −8 m/s.
a = −8 m/s ÷ 4 s = −2 m/s².

Answer: Acceleration −2 m/s²

A line that falls has a negative gradient. The cyclist is slowing down.

Now put the idea the other way round. If we know the acceleration, we can find the final velocity: v = u + at. A car at 5 m/s that accelerates at 2 m/s² for 4 s ends at 5 + 2 × 4 = 13 m/s.

A velocity time graph also gives distance. Take a steady velocity of 6 m/s for 10 s. The distance is 6 × 10 = 60 m. That is the area of the rectangle under the line: height 6 m/s times width 10 s.

The area between the line and the time axis is the displacement. While the velocity stays positive, the object never turns back, so the area is also the distance travelled. A triangle has area ½ × base × height.

A car speeds up, travels steadily, then brakes to rest. Each part has its own gradient and its own area.

Worked example

Find the distance the car travels in the 20 s of the graph above.

The distance is the area under the line. Split it into a triangle, a rectangle and a triangle.
From 0 s to 6 s the area is the triangle ½ × 6 × 12 = 36 m.
From 6 s to 16 s it is the rectangle 10 × 12 = 120 m.
From 16 s to 20 s it is the triangle ½ × 4 × 12 = 24 m.
Total distance = 36 m + 120 m + 24 m = 180 m.

Answer: 180 m

The same graph gives the accelerations. From 0 s to 6 s the gradient is 12 ÷ 6 = 2 m/s². From 6 s to 16 s the line is flat, so the acceleration is 0 m/s². From 16 s to 20 s the gradient is −12 ÷ 4 = −3 m/s².

When the acceleration is steady, the mean velocity is halfway between u and v. So the distance is ½ × (u + v) × t. This is the area of a trapezium. It works only when the acceleration is steady.

What if the velocity goes below zero? A trolley is pushed up a slope at 4 m/s. It slows, stops, then rolls back down. After 8 s it moves down the slope at 4 m/s. Take up the slope as positive.

The line crosses the time axis at 4 s. There the trolley stops for an instant, and then it moves back. The 5 s mark is hidden where the line crosses it.

The triangle above the axis is the trolley going up the slope: ½ × 4 × 4 = 8 m. The triangle below is it coming back: 8 m the other way. The displacement is 8 m − 8 m = 0 m. The trolley is back at its start.

The distance travelled is 8 m + 8 m = 16 m. So the area under a velocity time graph is the distance only while the velocity stays positive. Once it goes negative, the area below the axis is motion the other way.

The acceleration is steady all the time: (−4 − 4) ÷ 8 = −1 m/s². The trolley slows to a stop and speeds up the other way without any change in the acceleration.

Try it

Roll a ball down a slope

You need: a smooth board or flat plank, about 1.5 m long, a book to raise one end, a ball that rolls straight, small stones or chalk to mark places, a friend who can clap or tap a steady beat, about two each second.

Be safe: Do this on the floor, away from stairs and from people walking by. Keep the ball and stones away from small children.

Lay the board on the floor. Raise one end about 10 cm on the book. This is a gentle slope.
Hold the ball at the top. Your friend starts the beat. Let go on the first beat.
Roll it four times. On the first roll, mark where the ball is one beat after you let go. On the second roll, mark it two beats after, and so on.
Measure the gaps between the marks, one after another.
Do the whole set twice more and compare.

What you will see: Each gap is bigger than the one before. The gaps grow by about the same step each time. The ball covers more distance in each equal time, so its velocity is growing steadily. That is a steady acceleration.

If you cannot try it, this is what happens.

A wrong idea many people have

Acceleration is how fast something is going.

Why it feels right: In everyday speech, "accelerating" and "going fast" go together, and a fast car feels as if it has a lot of acceleration.

The test that breaks it: A car drives at a steady 30 m/s on a straight road. Its velocity does not change, so (v − u) ÷ t = 0 ÷ t = 0.

What is true: Acceleration is how quickly the velocity changes. A car at a very high, steady velocity has no acceleration. A car that starts from rest can have a large one.

A wrong idea many people have

A negative acceleration always means slowing down.

Why it feels right: "Deceleration" and a minus sign both sound like slowing, and in many first examples the sign is negative when something slows.

The test that breaks it: Take right as positive. A trolley moving left goes from −2 m/s to −8 m/s in 3 s. Its acceleration is (−8 − (−2)) ÷ 3 = −2 m/s². It is moving faster than before.

What is true: A negative acceleration points the negative way. If the object is moving the negative way too, it speeds up. It slows down only when the acceleration points against the velocity.

A wrong idea many people have

On a velocity time graph the height is the distance, and a flat line means the object is stopped.

Why it feels right: A distance time graph works that way, and the two graphs look alike.

The test that breaks it: Draw a flat line at 6 m/s for 10 s. The object moves 6 × 10 = 60 m, not 6 m. A stopped object is a line along the time axis, at 0 m/s.

What is true: On a velocity time graph the height is the velocity and the gradient is the acceleration. The distance is the area. A flat line above the axis is a steady velocity.

Key words

velocity: speed in a stated direction. It is a vector, in m/s.
acceleration: the change in velocity divided by the time taken, in m/s².
gradient: the steepness of a line: the rise divided by the run.
velocity time graph: a graph with time across and velocity up. The gradient is the acceleration and the area is the displacement.

Back to your guess

The answer is A: By how much its velocity changes each second.

How quickly the velocity changes is the acceleration. A bus that gains 2 m/s every second has an acceleration of 2 m/s². The distance and the top speed do not tell us that.

Keep it

Copy this into your exercise book.

Acceleration = (v − u) ÷ t, in m/s². On a velocity time graph the gradient is the acceleration and the area is the distance, while velocity stays positive.

Draw: A velocity time graph that rises, runs flat and falls, with the area split into a triangle, a rectangle and a triangle.