S4.8.8 Chapter 8

Stopping distances

A child steps into the road. Why can a car not stop the moment the driver sees her?

Your guess

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

Your guess

A The brakes take a long time before they start to work.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The car keeps moving while the driver reacts, and again while the brakes slow it.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Only the road decides how far a car goes before it stops.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Explain what affects thinking distance and braking distance
Use a measured reaction time to work out thinking distance

A driver sees danger, thinks, and presses the brake. The car then slows and stops. Both parts take time, and the car keeps moving for all of it.

The thinking distance is how far the car travels while the driver reacts, before the brakes start to work. The braking distance is how far it travels from the moment the brakes act until it stops.

The stopping distance is the two added together: stopping distance = thinking distance + braking distance.

See it three ways

As a picture

In words

The car travels on while the driver reacts. Then it travels on while the brakes work. The two distances add up to the stopping distance.

In symbols

10.5 m + 22.5 m = 33 m

The reaction time is the time from seeing the danger to pressing the brake. During it the car keeps its speed. So thinking distance = speed × reaction time.

Speeds on roads are often in km/h. For this sum, change them to m/s. One km/h is 1 000 m in 3 600 s, so divide by 3.6. A car at 72 km/h is at 72 ÷ 3.6 = 20 m/s.

How long is a reaction time? A simple test, such as catching a falling ruler, gives about 0.15 s to 0.25 s. A driver has to see, decide and act, and this takes longer, often about 0.7 s.

You can measure your own simple reaction time with a ruler. Your friend holds the ruler upright and drops it. You catch it. The ruler falls from rest with the acceleration g, 9.8 m/s². The distance it falls tells you the time.

On a velocity time graph the ruler’s line is v = g × t. The distance fallen is the area of the triangle: d = ½ × t × (g × t) = ½ × g × t². Turn this round to get t = √(2d ÷ g).

Worked example

A ruler falls 20 cm before Hana catches it. Find her reaction time. Use g = 9.8 m/s².

The ruler falls from rest, so d = ½ × g × t², and so t = √(2d ÷ g).
Change the distance to metres: 20 cm = 0.2 m.
t = √(2 × 0.2 ÷ 9.8) = √0.0408 ≈ 0.20 s.

Answer: 0.2 s

After 0.2 s the ruler moves at 1.96 m/s. The triangle has area ½ × 0.2 × 1.96 = 0.196 m, about 20 cm.

Try it

Catch a falling ruler

You need: a ruler, or a long strip of card marked in centimetres, a friend.

Be safe: Let the ruler drop. Do not throw it. Keep your face away from the falling end.

Your friend holds the ruler upright by its top end. Hold your thumb and finger open, either side of the bottom end, with the 0 mark level with them.
Your friend lets go without warning. Close your fingers as soon as you see it fall.
Read the mark at the top of your thumb. This is the distance the ruler fell, in centimetres.
Do it five times and take the usual value. Look up the time in the table.

What you will see: Most people catch the ruler after 10 cm to 30 cm. That is a time of about 0.14 s to 0.25 s. A driver reacts more slowly, because there is more to decide than a ruler falling.

If you cannot try it, this is what happens.

Now use a reaction time for a driver. Thinking distance = speed × reaction time. At twice the speed, the car goes twice as far in the same time. So thinking distance grows in proportion to speed.

Worked example

Lucía drives at 90 km/h. Her reaction time is measured as 0.8 s. Find her thinking distance.

Change the speed to m/s: 90 km/h ÷ 3.6 = 25 m/s.
The car keeps this speed while she reacts, so thinking distance = speed × time.
Thinking distance = 25 m/s × 0.8 s = 20 m.

Answer: 20 m

The car travels 20 m before the brakes start to act.

Several things make the reaction time longer. These are tiredness, alcohol, some medicines and drugs, and distractions such as a phone. A longer reaction time means a longer thinking distance, at any speed.

Now the braking distance. The brakes give a braking force, and by the second law a = F ÷ m. We call a steady slowing a deceleration. Take a car that slows by 5 m/s every second, a deceleration of 5 m/s².

From 20 m/s the car needs 20 ÷ 5 = 4 s to stop. The graph is a straight line down. So the braking distance is the area of a triangle: ½ × 4 × 20 = 40 m.

Now try 10 m/s, then 30 m/s, with the same deceleration. From 10 m/s the car stops in 2 s. From 30 m/s it takes 6 s. The graph shows the three triangles.

Same deceleration, so the same gradient. A higher starting speed makes a taller and wider triangle.

The areas are ½ × 2 × 10 = 10 m, ½ × 4 × 20 = 40 m and ½ × 6 × 30 = 90 m. Double the speed from 10 to 20 m/s, and the braking distance goes from 10 m to 40 m. It is four times as big, because both the base and the height of the triangle double.

So the braking distance grows with the square of the speed. This holds whenever the deceleration is the same at every starting speed.

Worked example

A car travels at 72 km/h. The driver’s reaction time is 0.7 s. The car brakes with a deceleration of 5 m/s². Find the stopping distance.

Change the speed to m/s: 72 km/h ÷ 3.6 = 20 m/s.
Thinking distance = 20 m/s × 0.7 s = 14 m.
The car needs 20 ÷ 5 = 4 s to stop, so the braking distance is ½ × 4 × 20 = 40 m.
Stopping distance = 14 m + 40 m = 54 m.

Answer: 54 m

The stopping distance is the whole bar, 54 m.

The bars below show the three speeds. The thinking part grows in step with the speed. The braking part grows much faster.

With a reaction time of 0.7 s and a deceleration of 5 m/s². Thinking distances are 7, 14 and 21 m. Braking distances are 10, 40 and 90 m.

What changes the braking distance? First, the speed, as shown. Second, the grip between tyres and road. Third, the condition of the brakes and tyres. Fourth, the mass of the vehicle. Fifth, the slope of the road.

On a wet or icy road the tyres grip less. The braking force is smaller, so the deceleration is smaller, and the car takes longer to stop. Worn tyres and worn brakes do the same.

With the same braking force, a heavier vehicle has a smaller deceleration, since a = F ÷ m. So it takes longer to stop. This is why trucks have much bigger brakes. Going downhill also makes the braking distance longer.

Take the 20 m/s car again. Suppose the road is wet and the deceleration is only 2.5 m/s², half as much. The car needs 8 s to stop, and the braking distance is ½ × 8 × 20 = 80 m. This is twice the dry distance. The values are round numbers for the example.

The thinking distance is the same. The wet road doubles the braking distance.

Note which part each cause changes. Tiredness, alcohol and a phone change the thinking distance. Wet roads, worn tyres, a heavy load and a slope change the braking distance. Speed changes both.

A wrong idea many people have

The stopping distance is how far the car goes after the brakes are pressed.

Why it feels right: We usually think of stopping as the part we feel, with the brakes on and the car slowing.

The test that breaks it: A car at 20 m/s and a reaction time of 0.7 s travels 14 m before the brakes do anything. A person 45 m away is beyond the braking distance of 40 m. Yet the car travels 14 m + 40 m = 54 m, so it reaches them.

What is true: The stopping distance is the thinking distance plus the braking distance. The car moves for the whole of the thinking time, with no braking.

A wrong idea many people have

If the speed doubles, the stopping distance doubles.

Why it feels right: Distance and speed go together, and thinking distance does double with speed.

The test that breaks it: From 10 m/s to 20 m/s the thinking distance goes from 7 m to 14 m, which is double. The braking distance goes from 10 m to 40 m, which is four times. The stopping distance goes from 17 m to 54 m, about three times.

What is true: Thinking distance is in proportion to speed. Braking distance grows with the square of the speed. So the stopping distance grows faster than the speed.

A wrong idea many people have

A tired driver needs a longer braking distance.

Why it feels right: A tired driver stops later, so it feels as if the braking has gone wrong.

The test that breaks it: Take a car at 20 m/s. With a reaction time of 0.7 s the thinking distance is 14 m. With 1.5 s it is 30 m. The braking distance is 40 m in both cases, since the brakes and the road are the same.

What is true: Reaction time changes only the thinking distance. The braking distance depends on the speed, the grip, the brakes and the load.

Key words

reaction time: the time from seeing danger to pressing the brake.
thinking distance: the distance the car travels during the reaction time. It is speed × reaction time.
braking distance: the distance the car travels while the brakes slow it to rest.
stopping distance: the thinking distance plus the braking distance.

Back to your guess

The answer is B: The car keeps moving while the driver reacts, and again while the brakes slow it.

There are two stretches. First the car travels on while the driver reacts. Then it travels on while the brakes slow it. The two distances add up to the stopping distance.

Keep it

Copy this into your exercise book.

Stopping distance = thinking distance + braking distance. Thinking distance = speed × reaction time. Braking distance grows with the square of the speed. Wet roads and heavy loads make it longer.

Draw: A bar split into thinking and braking parts, and a velocity time graph that falls to zero.