Linear Momentum and Newton's Second Law

1Which is harder to stop?

A football and a wheelbarrow full of bricks both roll towards you down a gentle slope at walking pace. You would happily stop the football with your foot, but you would jump out of the way of the wheelbarrow. Now picture a bus creeping forward at $2\mathrm{m}{\mathrm{s}}^{-1}2\backslash \; \backslash mathrm\{m\backslash ,s^\{-1\}\}$ in a Nyabugogo traffic jam and a moto speeding past it at $15\mathrm{m}{\mathrm{s}}^{-1}15\backslash \; \backslash mathrm\{m\backslash ,s^\{-1\}\}$. Which one would need the larger force, or the longer time, to stop?

How hard a moving body is to stop depends on both its mass and its velocity. Physicists combine the two into one quantity: momentum.

2Experiment 1: the same push for the same time

You need: a trolley, three 0.50 kg masses (or bags of sand), a spring balance reading to 0.1 N, a stopwatch, metre rules and a phone that can record video.

1. Load the trolley so that its total mass is 0.50 kg. Lay metre rules along a smooth, level floor.
2. Pull the trolley from rest with the spring balance, keeping the reading at a steady 0.50 N, for exactly 2.0 s. A partner calls out when 2.0 s have passed and you let go.
3. Film the trolley beside the rules. From the video, find how far it moves in the first 0.20 s after you let go, and so its speed $vv$.
4. Repeat with total masses of 1.0 kg and 1.5 kg, using the same force and the same time.

Figure 1.1: The same force acts for the same time on trolleys of different mass.

These are illustrative results. The heavier the trolley, the slower it ends up, but the product $mvmv$ is almost the same each time, and almost equal to force × time. (It is a little smaller because friction also acts.) The same force acting for the same time always gives a body the same momentum, whatever its mass.

3Momentum has a direction

Because momentum is a vector, we choose one direction as positive. A body moving the opposite way has a negative momentum.

Figure 1.2: Momentum carries the sign of the velocity. Two bodies can have momenta of the same size that cancel when they are added.

The change in momentum of a body is its final momentum minus its initial momentum, $\mathrm{\Delta }p=mv-mu\backslash Delta\; p\; =\; mv\; -\; mu$, where $uu$ is the initial velocity and $vv$ the final velocity. Always put in the signs. A ball that bounces back has a larger change in momentum than one that simply stops (Worked example 2).

4Newton's second law, in terms of momentum

In S1 you met Newton's second law as $F=maF\; =\; ma$. Put $a=\frac{v-u}{t}a\; =\; \backslash frac\{v\; -\; u\}\{t\}$ into it:

This is the form Newton himself used. It is more general than $F=maF\; =\; ma$: it still works when the mass changes, as it does for a rocket burning fuel.