Impulse and Change in Momentum

1Why does the pestle crush the grain?

A woman pounds dried cassava in a wooden mortar. She lifts the heavy pestle and brings it down: when it strikes the hard pieces of cassava it stops almost instantly, and they are crushed. If a cloth were folded over the grain, the same pestle moving at the same speed would stop more gently and crush much less.

In both cases the pestle loses the same momentum. What changes is the time it takes to lose it. In S1 you saw that a longer stopping time needs a smaller force. In this lesson that idea gets a name, impulse, and becomes one of the most useful tools in road safety.

2Experiment 2: catching a water balloon

You need: small balloons filled with water, and a partner. Do this outdoors.

1. Stand 3 m apart. Your partner throws a balloon gently underarm to you.
2. Catch it with stiff arms and hands held still.
3. Catch another balloon, thrown the same way, but this time move your hands back towards your body as you catch it, bringing it to rest gradually.
4. Repeat each catch three times and count how many balloons burst.

Observation: balloons caught with stiff hands often burst; balloons caught with "giving" hands rarely do.

Explanation: each balloon arrives with the same momentum and ends at rest, so its change in momentum is the same. Moving your hands back makes the stopping time longer, so the force on the balloon is smaller, too small to burst it.

3Impulse and the change in momentum

Start from Newton's second law in its momentum form, $F=\frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}F\; =\; \backslash frac\{\backslash Delta\; p\}\{\backslash Delta\; t\}$, and multiply both sides by $\mathrm{\Delta }t\backslash Delta\; t$:

The product of a force and the time for which it acts is called the impulse. The equation says that the impulse on a body equals its change in momentum. For a fixed change in momentum, a long time means a small force, and a short time means a large force.

4Impulse from a force–time graph

In a real collision the force is not constant: it rises from zero to a peak and falls back. The impulse is then the area under the force–time graph, and the "average force" is the constant force that would give the same area in the same time.

Figure 2.1: The same ball stopped in two ways. Both triangles have the same area, $\frac{1}{2}\times 400\text{ N}\times 0.020\text{ s}=\frac{1}{2}\times 100\text{ N}\times 0.080\text{ s}=4.0\text{ N s}\backslash frac\{1\}\{2\}\; \backslash times\; 400\backslash \; \backslash text\{N\}\; \backslash times\; 0.020\backslash \; \backslash text\{s\}\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 100\backslash \; \backslash text\{N\}\; \backslash times\; 0.080\backslash \; \backslash text\{s\}\; =\; 4.0\backslash \; \backslash text\{N\; s\}$, because the change in momentum is the same. Making the time four times longer makes the peak force four times smaller.

5Road safety: impulse saves lives

In a crash, a person's momentum must fall to zero; nothing can change that. Every safety device works by making the stopping time longer, so the force on the body is smaller.

• Seat belts hold you to the car, which is stopping over the time its front crumples, and stretch slightly. Without a belt, you keep moving until you hit the dashboard or windscreen, which stops you in a much shorter time.
• Crumple zones are the front and back of a car, designed to fold up slowly in a crash, which lengthens the time for the passenger compartment to stop.
• Airbags inflate in a fraction of a second and then let gas out as your head presses into them, so your head stops over a longer time and over a larger area.
• Helmets have a hard shell that spreads the blow over a large area, and a thick foam liner that crushes, so the head stops over a longer time.

Figure 2.2: Inside a motorcycle helmet. A helmet that has already taken a hard blow must be replaced, even if it looks fine: its foam has been crushed and cannot absorb a second impact.

The World Health Organization reports that correct helmet use can reduce the risk of death in a crash by more than 6 times and the risk of brain injury by up to 74%, and that wearing a seat belt can reduce the risk of death among vehicle occupants by up to 50% (WHO, Road traffic injuries fact sheet, 2026). Road traffic injuries are the leading cause of death for children and young people aged 5 to 29 worldwide.