The second law, momentum and impulse

Momentum: mass and velocity together

Narrated video. Transcript:

A tennis ball and a cricket ball come at you at the same speed. Which would you rather stop? The cricket ball hurts more because it has more mass.

Now two identical cricket balls: one gently tossed, one from a fast bowler at 40 metres per second. The faster one is much harder to stop.

Both mass and velocity matter. Newton combined them into one quantity, momentum: p equals m times v.

Momentum is a vector. It points the same way as the velocity. Its unit is kilogram metre per second.

Watch a loaded truck and an auto rickshaw roll at the same speed. The truck has far more momentum, so the same brakes take much longer to stop it.

A force changes momentum. Push for a longer time and the momentum changes more. Push harder and it changes faster.

Force can also change only the direction. A stone whirled on a string keeps its speed, but the string keeps turning its momentum round.

So force is about how fast momentum changes. That is Newton’s second law.

F = ma and impulse

Narrated video. Transcript:

The second law says: the rate of change of momentum of a body is proportional to the applied force, and happens in the direction of the force.

With the right choice of units, F equals d p by d t. For a fixed mass, d p by d t is m times d v by d t, which is m a. So F equals m a.

The unit of force is the newton. One newton gives a one kilogram mass an acceleration of one metre per second squared.

F equals m a is a vector equation. Each component works on its own: F x equals m a x, F y equals m a y. And if F is zero, a is zero, which is just the first law again.

Now a quick hit, like a bat striking a ball. The force is large and lasts a very short time. The product F times delta t is called impulse.

Impulse equals the change in momentum. On a force–time graph, impulse is the area under the curve. Watch the area shade in.

A fielder catching a cricket ball must remove all its momentum. That is a fixed impulse. Stop it in a hundredth of a second and the force is huge.

So the fielder pulls the hands back. The same area spreads out over a longer time, so the graph is lower: a smaller force and no stinging palms.

The same idea is behind airbags, sand pits for long jump, and packing glassware in straw.

Try it: catch a cricket ball

Interactive simulation. Things to try:

• Press Bowl the ball and watch the force–time graph fill in.
• Press Keep for comparison, then increase the stopping time and bowl again.
• The shaded area (impulse) stays the same for the same ball speed. Only the height (force) changes.

Quick check

A batsman hits a 0.15 kg ball straight back towards the bowler. The ball arrived at 12 m/s and leaves at 12 m/s. What is the size of the impulse given to the ball?

1. 0 N·s, because the speed does not change
2. 1.8 N·s
3. 3.6 N·s
4. 0.9 N·s

Why pull the hands back?

A fielder moves the hands backwards while catching a fast ball. Why does this hurt less?

1. It reduces the momentum of the ball before it is caught
2. It reduces the impulse the hands must give the ball
3. It increases the time to stop the ball, so the average force is smaller
4. It increases the mass of the hands

Summary

• Momentum: p = mv
• Second law: F = dp/dt = ma
• Impulse: Impulse = F·Δt = Δp
• Unit of force: 1 N = 1 kg·m s⁻²

• Momentum p = mv is a vector, in the direction of the velocity.
• Force equals the rate of change of momentum. For a fixed mass this is F = ma, and it holds for each component separately.
• Impulse F Δt equals the change in momentum. It is the area under a force–time graph.
• For the same change in momentum, a longer time means a smaller force. That is why fielders pull their hands back.