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Conservation of momentum

Rockets and landing · level 6, lesson 1 of 6

A question to start

Two astronauts float at rest in the space station, facing each other. They push off each other's hands. One has a mass of 60 kg and the other 90 kg. Who drifts away faster, and how much faster?

The idea

When two bodies push on each other, the third law says the forces are equal and opposite, and they act for the same time. So the impulses are equal and opposite: whatever momentum one gains, the other loses. Added together, nothing changes.

Conservation of momentum: if no outside force acts on a group of bodies, their total momentum stays the same.

For two bodies:

\[m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2,\]

with \(u\) the velocities before and \(v\) after (signs show direction).

The astronauts. Total momentum before: 0. After: \(60 v_1 + 90 v_2 = 0\), so \(v_1 = -1.5 v_2\). The lighter one moves 1.5 times as fast, the opposite way. If the heavier drifts at 0.4 m/s, the lighter goes at 0.6 m/s.

A rocket and its exhaust are a group with no outside force (in space). Start at rest: total momentum 0. When the rocket throws exhaust backwards, the rocket must gain momentum forwards so that the total stays 0. That is all a rocket does.

Docking. A 10,000 kg capsule moving at 0.1 m/s docks with a 400,000 kg station at rest. Afterwards they move together: \(10{,}000 \times 0.1 = 410{,}000\,v\), so \(v = 0.0024\) m/s. The station barely notices.

Worked examples

1. Skaters. A 50 kg skater at 4 m/s runs into and holds onto a 70 kg skater at rest. \(200 = 120\,v\), so they glide on at 1.67 m/s.

2. A sample capsule. A spacecraft of 1,000 kg at rest throws a 20 kg capsule away at 2 m/s. \(0 = 20 \times 2 + 980\,v\), so \(v = -0.041\) m/s: the craft drifts back at 4 cm/s.

3. Two rovers. A 900 kg rover at 0.04 m/s nudges a 185 kg rover at rest, and they move together. \(36 = 1085\,v\), so \(v = 0.033\) m/s.

Try it

  1. A 70 kg astronaut and a 30 kg crate float at rest. The astronaut pushes the crate away at 1.4 m/s. Find the astronaut's velocity.
  2. A 2 kg ball at 3 m/s hits a 1 kg ball at rest. Afterwards the 2 kg ball moves at 1 m/s in the same direction. Find the other ball's speed.
  3. Why does momentum stay the same, even though each body's momentum changes?
  4. A ship's engines are off. A crew member walks from the back of the ship to the front. Does the ship move? Does the group's centre of mass?
Answers
  1. \(0 = 30 \times 1.4 + 70\,v\), so \(v = -0.6\) m/s: 0.6 m/s the opposite way.
  2. \(6 = 2 + 1\,v\), so \(v = 4\) m/s.
  3. The forces between them are equal and opposite and act for the same time, so the changes cancel.
  4. The ship moves backwards a little while they walk, and stops when they stop. The centre of mass of the ship and crew does not move.

Common mistake

Forgetting directions. Momentum is a vector. In "the astronauts push apart", one velocity is negative; adding speeds without signs gives a total that is not zero, and the wrong answer.

Where it leads