Linear momentum and its conservation

AS Level Physics · Dynamics. 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 principle of conservation of momentum states that the total momentum of a system stays constant, provided no resultant external force acts on it. It holds for every collision and every explosion. Momentum is a vector, so choose a positive direction and give a minus sign to anything moving the other way.

Kinetic energy is different. In an elastic collision the total kinetic energy is also conserved. In an inelastic collision some kinetic energy is transferred to other forms, although total energy is still conserved. Objects that stick together always collide inelastically.

In two dimensions, momentum is conserved separately in each of two perpendicular directions.

Rules to remember

Common mistake

Adding the momenta of objects moving in opposite directions as if they were all positive. One direction must be negative, and a rebound changes the sign of the velocity.

Worked example

A 0.40 kg trolley moving right at 5.0 m s−1 collides head-on with a 0.60 kg trolley moving left at 2.0 m s−1. The 0.40 kg trolley rebounds to the left at 1.0 m s−1. What is the velocity of the 0.60 kg trolley, and is the collision elastic?

  1. Take right as positive. Momentum before = (0.40 × 5.0) + (0.60 × −2.0) = 2.0 − 1.2 = 0.80 kg m s−1.
  2. Momentum after = (0.40 × −1.0) + 0.60v = 0.80, so 0.60v = 1.20 and v = 2.0 m s−1.
  3. Relative speed of approach = 5.0 + 2.0 = 7.0 m s−1; relative speed of separation = 2.0 + 1.0 = 3.0 m s−1.
  4. These are not equal, so kinetic energy is not conserved.

Answer: 2.0 m s−1 to the right; the collision is inelastic.

Practice questions

Try each one, then open the answer.

1. A ball of mass 0.20 kg hits a wall at 6.0 m/s and rebounds at 4.0 m/s. The contact time is 0.050 s. What is the average force on the ball?

  1. A
    8.0 N
  2. B
    16 N
  3. C
    24 N
  4. D
    40 N
Show answer

Answer: D. Δp = 0.20 × (6.0 − (−4.0)) = 2.0 kg m/s. F = Δp/Δt = 2.0 ÷ 0.050 = 40 N.

2. Two identical balls travelling at equal speeds in opposite directions collide head-on and stop. Which statement is correct?

  1. A
    momentum was not conserved
  2. B
    the collision is perfectly elastic
  3. C
    the total momentum before and after is zero, and the collision is inelastic
  4. D
    kinetic energy was conserved
Show answer

Answer: C. Equal and opposite momenta sum to zero before and after. All the kinetic energy is lost, so it is perfectly inelastic.

3. Sphere P moving at 5.0 m s−1 collides elastically head-on with sphere Q, which is moving at 2.0 m s−1 in the same direction. After the collision P moves at 3.0 m s−1 in its original direction. What is the velocity of Q?

  1. A
    4.0 m s−1
  2. B
    5.0 m s−1
  3. C
    6.0 m s−1
  4. D
    8.0 m s−1
Show answer

Answer: C. In an elastic collision the relative speed of approach (5.0 − 2.0 = 3.0 m s−1) equals the relative speed of separation, so vQ − 3.0 = 3.0 and vQ = 6.0 m s−1.

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