Newton's second law, momentum and impulse

A force changes the motion

The first law says what happens with no net force. The second law says what a force does: it changes the velocity. A bigger force changes it faster; a bigger mass resists more.

Momentum

The momentum of a body is its mass times its velocity:

It is a vector, pointing the way the body moves, measured in $\text{kg m/s}$. A slow lorry and a fast bullet can have similar momenta: one has a large mass, the other a large speed.

Newton's second law

The net force on a body equals the rate of change of its momentum:

For a body whose mass does not change, $\Delta p = m\,\Delta v$, so

The force and the acceleration point the same way. One newton is the force that gives a mass of $1\ \text{kg}$ an acceleration of $1\ \text{m/s}^2$: $1\ \text{N} = 1\ \text{kg m s}^{-2}$.

Watch out. $F$ is the net force. If you push a box with $50\ \text{N}$ and friction pulls back with $20\ \text{N}$, use $30\ \text{N}$ in $F = ma$.

Impulse

A bat hits a ball for a few milliseconds with a huge, changing force. We cannot follow the force, but we can measure its total effect, the impulse:

On a force-time graph, the impulse is the area under the curve. The same impulse can come from a big force for a short time or a small force for a long time. See it.

You should find that stretching the time from $0.05\ \text{s}$ to $0.2\ \text{s}$ cuts the force to a quarter. That is why a goalkeeper's gloves are padded and why you bend your knees when you land.

A worked example

A hockey ball of mass $0.16\ \text{kg}$ comes at $14\ \text{m/s}$ and is hit straight back at $22\ \text{m/s}$. The stick touches it for $0.012\ \text{s}$. Find the impulse and the average force.

1. Choose a positive direction: the way the ball goes after the hit.
2. Change of momentum. $0.16\ \text{kg} \times 22\ \text{m/s} - 0.16\ \text{kg} \times (-14\ \text{m/s}) = 3.52\ \text{kg m/s} + 2.24\ \text{kg m/s} = 5.76\ \text{kg m/s}$.
3. Impulse $= 5.76\ \text{N s}$. Average force $= \dfrac{5.76\ \text{kg m/s}}{0.012\ \text{s}} = 480\ \text{N}$.

Think first. Why add the two momenta? The ball reverses, so its velocity changes by $14\ \text{m/s} + 22\ \text{m/s} = 36\ \text{m/s}$, not by $22 - 14$.

Practise with new numbers

Try it in Python

The program stops the same football in four different times and finds the average force each time.

Remember these

• Momentum $p = mv$, a vector, in $\text{kg m/s}$.
• Second law: $F = \dfrac{\Delta p}{\Delta t}$, which is $F = ma$ for a fixed mass. Use the net force.
• $1\ \text{N} = 1\ \text{kg m s}^{-2}$.
• Impulse $= F\Delta t = \Delta p$ = the area under a force-time graph.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A net force of $12\ \text{N}$ acts on a $3.0\ \text{kg}$ trolley. What is its acceleration?

2. A $70\ \text{kg}$ runner moves at $8\ \text{m/s}$. What is her momentum?

3. A force-time graph is a rectangle, $200\ \text{N}$ high and $0.05\ \text{s}$ wide. What impulse does it give?

4. Why does a glass break when it falls on a floor but not on a thick carpet?