Momentum and its conservation

Momentum

p = mv. Momentum is a vector measured in kg m/s. Its direction is the direction of the velocity, so give one direction a positive sign and the other negative.

Conservation

In any interaction, the total momentum of a system stays the same, provided no external resultant force acts:

total momentum before = total momentum after.

Using it

• Collisions: m1u1 + m2u2 = m1v1 + m2v2. If they stick together, use (m1 + m2)v.
• Explosions (things pushing apart from rest): total momentum before is zero, so afterwards the momenta are equal and opposite.
• A change of direction counts: a ball bouncing straight back changes momentum by m(v - u) with signs, which can be twice mv.

Worked example

A 2.0 kg trolley at 3.0 m/s hits a stationary 1.0 kg trolley and they stick together. Find their common velocity.

1. Before: 2.0 x 3.0 + 1.0 x 0 = 6.0 kg m/s
2. After: (2.0 + 1.0)v = 6.0

Answer: v = 2.0 m/s in the original direction

Check yourself

1. Find the momentum of a 0.16 kg ball moving at 25 m/s, in kg m/s.
   Show answer

   4 kg m/s. p = mv = 0.16 x 25 = 4.0 kg m/s.

2. A 3.0 kg trolley at 4.0 m/s collides with a stationary 1.0 kg trolley and they stick together. Find their velocity in m/s.
   Show answer

   3 m/s. 12 = 4.0v, so v = 3.0 m/s.

3. A 4.0 kg rifle fires a 0.010 kg bullet at 400 m/s. Find the recoil speed of the rifle in m/s.
   Show answer

   1 m/s. Total momentum is zero: 4.0v = 0.010 x 400 = 4.0, so v = 1.0 m/s backwards.

4. A 2.0 kg ball moving at +5.0 m/s hits a 3.0 kg ball moving at -2.0 m/s. They stick together. Find their velocity in m/s (positive means the first ball's original direction).
   Show answer

   0.8 m/s. Total p = 10 - 6.0 = 4.0 kg m/s; v = 4.0 / 5.0 = +0.80 m/s.

5. A 0.050 kg ball hits a wall at 4.0 m/s and rebounds at 4.0 m/s. Find the size of its change in momentum, in kg m/s.
   Show answer

   0.4 kg m/s. Δp = m(v - u) = 0.050 x (4.0 - (-4.0)) = 0.40 kg m/s.

6. When is the total momentum of a system conserved?
   • Only in collisions where objects stick together
   • Only when kinetic energy is conserved
   • Only when the objects are at rest
   • When no external resultant force acts on it
   Show answer

   When no external resultant force acts on it. Momentum is conserved in all interactions if there is no external resultant force.

7. Why must you use signs in momentum calculations?
   • Mass can be negative
   • Momentum is a vector, so direction matters
   • Speed can be negative
   • Momentum is always conserved
   Show answer

   Momentum is a vector, so direction matters. Velocities in opposite directions have opposite signs, so their momenta partly cancel.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.