Momentum

AS Level Mathematics · Mechanics. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

The momentum of a particle is its mass multiplied by its velocity. Because velocity has a direction, momentum is a vector. In a straight line problem, choose a positive direction first. Anything moving the other way has negative velocity and negative momentum.

When two bodies collide directly, the forces between them are equal and opposite, so the total momentum of the two bodies is the same before and after the impact. This is conservation of linear momentum.

If the bodies coalesce (join together), they move as one body with the combined mass and a single common velocity.

Rules to remember

Common mistake

Forgetting the minus sign for a body moving in the opposite direction. Draw a before and after diagram with one positive direction and put a sign on every velocity.

Worked example

A of mass 4 kg moving at 3 m s−1 and B of mass 1 kg moving at 2 m s−1 travel towards each other, collide and coalesce. Find their common velocity.

  1. Take the direction of A as positive, so the velocity of B is −2 m s−1.
  2. Momentum before = 4 × 3 + 1 × (−2) = 12 − 2 = 10 kg m s−1.
  3. Momentum after = (4 + 1)v = 5v, so 5v = 10 and v = 2.

Answer: 2 m s−1 in the original direction of A

Practice questions

Try each one, then open the answer.

1. A of mass 3 kg moving at 4 m s−1 and B of mass 2 kg moving at 3 m s−1 travel towards each other and collide. After the collision A continues in its original direction at 1 m s−1. Find the velocity of B.

  1. A
    1.5 m s−1 in the direction B was moving
  2. B
    4.5 m s−1 in the direction A was moving
  3. C
    1.5 m s−1 in the direction A was moving
  4. D
    3 m s−1 in the direction B was moving
Show answer

Answer: C. Take the direction of A as positive: 3(4) + 2(−3) = 3(1) + 2v, so 6 = 3 + 2v and v = +1.5. Positive means B now moves in the direction A was moving, i.e. B has been reversed.

2. A particle of mass 2 kg moving at 5 m s−1 collides with a particle of mass 3 kg at rest. They coalesce. Find their common speed after the collision.

  1. A
    2 m s−1
  2. B
    5 m s−1
  3. C
    3.33 m s−1
  4. D
    1.5 m s−1
Show answer

Answer: A. Conservation of momentum: 2 × 5 + 3 × 0 = (2 + 3)v, so v = 10/5 = 2 m s−1. Divide by the combined mass, not just one of them.

3. A particle of mass 0.3 kg moving at 4 m s−1 is struck so that it moves back along the same line. The magnitude of the change in its momentum is 2.1 kg m s−1. Find its new speed.

  1. A
    11 m s−1
  2. B
    3 m s−1
  3. C
    7 m s−1
  4. D
    4 m s−1
Show answer

Answer: B. Take the new direction as positive. Initial momentum = 0.3 × (−4) = −1.2 and final momentum = 0.3v, so 0.3v + 1.2 = 2.1, giving v = 3 m s−1. Ignoring the reversal gives 0.3v − 1.2 = 2.1 and the wrong 11.

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