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Momentum

Rockets and landing · level 5, lesson 3 of 6

A question to start

The Perseverance rover drives at about 4 cm per second. A rifle bullet flies at about 900 m/s. Which is harder to stop?

The idea

How hard something is to stop depends on both its mass and its speed. Newton called the product the quantity of motion; we call it momentum:

\[p = m v.\]

Its unit is kg·m/s. Momentum has a direction, like velocity: a rover going east and one going west have opposite momenta.

Mass Speed Momentum
Perseverance, at top speed 1,025 kg 0.042 m/s 43 kg·m/s
A rifle bullet 0.01 kg 900 m/s 9 kg·m/s
A 70 kg person, walking 70 kg 1.4 m/s 98 kg·m/s
A car on the motorway 1,500 kg 30 m/s 45,000 kg·m/s
A lander under its parachute 2,000 kg 90 m/s 180,000 kg·m/s
Saturn V exhaust, each second 13,000 kg about 2,600 m/s about 34,000,000 kg·m/s

The slow rover has nearly five times the momentum of the fast bullet.

Newton's second law, the momentum way. Newton actually wrote his second law as: force equals the rate of change of momentum,

\[F = \frac{\Delta p}{\Delta t}.\]

When the mass stays the same, \(\Delta p = m\Delta v\), and this is \(F = ma\) again. But it also works when mass changes, as it does for a rocket throwing out exhaust. The last row of the table shows why: each second the Saturn V's engines gave about 34 million kg·m/s of momentum to their exhaust, which is a force of 34 MN, the thrust.

Worked examples

1. The rover. \(p = 1025 \times 0.042 = 43\) kg·m/s.

2. Stopping a car. To stop a 45,000 kg·m/s car in 5 s, the brakes need \(F = 45{,}000/5 = 9{,}000\) N.

3. The exhaust. An engine throws out 300 kg of gas each second at 2,800 m/s. Momentum given to the gas per second: \(300 \times 2800 = 840{,}000\) kg·m/s, so the thrust is 840 kN. (That is about one Merlin engine.)

Try it

  1. Find the momentum of a 0.5 kg ball thrown at 20 m/s.
  2. A 2,000 kg lander falls at 90 m/s. A 10,000 kg one falls at 18 m/s. Which has more momentum?
  3. What force stops a 1,500 kg car from 20 m/s in 4 s?
  4. An engine throws out 100 kg of gas each second at 3,500 m/s. What is its thrust?
  5. Why is momentum more useful than speed alone for describing a crash?
Answers
  1. \(10\) kg·m/s.
  2. They are equal: \(180{,}000\) kg·m/s each.
  3. \(\Delta p = 30{,}000\); \(F = 7{,}500\) N.
  4. \(350{,}000\) N, or 350 kN.
  5. It includes the mass: a slow, heavy thing can be just as hard to stop as a fast, light one.

Common mistake

Thinking momentum and energy are the same. Momentum is \(mv\) and has a direction; kinetic energy is \(\tfrac12mv^2\) and does not. A bullet and a rover with the same momentum have very different energies: the bullet's is far larger.

Where it leads