Successive impacts

One collision at a time

1. Use CLM and NEL for the first impact.

2. Use those velocities as the starting values for the next impact.

3. Keep one positive direction throughout and record every velocity on a diagram.

Will there be another collision?

After all the impacts so far, compare velocities of neighbouring particles. A particle behind collides again only if it is moving faster in the same direction than the one in front (or they move towards each other). If the one behind is slower, at rest, or moving away, there are no more collisions.

Particle and wall

A particle that hits a wall rebounds at e times its speed and may then meet the other particle again. Write its velocity after the wall as negative (towards the other particle) and treat the next impact as a head-on collision.

Worked example

Particles A (1 kg), B (2 kg) and C (4 kg) lie at rest in a straight line. A is projected at 6 m/s towards B. For every impact e = 0.5. Find the velocities after A hits B and after B hits C, and show there are no more impacts.

1. A and B: 6 = v_A + 2v_B and v_B - v_A = 3, so v_B = 3 and v_A = 0.
2. B and C: 6 = 2v_B + 4v_C and v_C - v_B = 1.5, so v_C = 1.5 and v_B = 0.
3. A and B are both at rest and C moves away, so there are no further impacts.

Answer: A at rest, B at rest, C moving at 1.5 m/s

Check yourself

1. Repeat the worked example with e = 1 for every impact. Find the speed of C after B hits it, to 3 significant figures.
   Show answer

   2.67 m/s. A and B: v_A = -2, v_B = 4. B and C: 8 = 2v_B + 4v_C and v_C - v_B = 4, so v_C = 2.67 m/s.

2. In that e = 1 case, A moves at 2 m/s and B at 4/3 m/s, both away from C, and C moves at 8/3 m/s away from B. Is there another collision?
   • No: B is behind A in that direction but slower, and C moves away
   • Yes: A and B collide again
   • Yes: B and C collide again
   • It cannot be decided
   Show answer

   No: B is behind A in that direction but slower, and C moves away. Moving away from C, A is in front at 2 m/s and B follows at 4/3 m/s, so B never catches A.

3. P (2 kg) moving at 4 m/s hits Q (1 kg) at rest, e = 0.5. Q then hits a fixed wall at right angles, with e = 0.5. Find the speed of Q after hitting the wall.
   Show answer

   2 m/s. P and Q: v_P = 2, v_Q = 4. After the wall Q moves back at 0.5 × 4 = 2.00 m/s.

4. P is still moving towards the wall at 2 m/s and Q is moving towards P at 2 m/s. They collide again with e = 0.5. Find the speed of Q after this second collision.
   Show answer

   2 m/s. CLM (towards wall positive): 4 - 2 = 2v_P + v_Q. NEL: v_Q - v_P = 0.5 × 4 = 2. So 2 = 3v_P + 2, v_P = 0 and v_Q = 2 m/s.

5. A (1 kg) moving at 9 m/s hits B (2 kg) at rest, then B hits C (4 kg) at rest. e = 0.25 for both impacts. Find B's speed after hitting C, to 3 significant figures.
   Show answer

   0.625 m/s. A and B: v_A = 1.50, v_B = 3.75. B and C: v_B = 0.625 m/s, v_C = 1.56 m/s.

6. In the last question, A moves at 1.5 m/s and B at 0.625 m/s, in the same direction. What happens next?
   • There are no more collisions
   • B catches C
   • A catches B and they collide again
   • A and C collide directly
   Show answer

   A catches B and they collide again. A is behind B and faster in the same direction, so it catches up.

7. How do you decide whether two particles moving in the same direction will collide again?
   • The one behind must be moving faster than the one in front
   • The heavier one must be behind
   • The one in front must be faster
   • They always collide again
   Show answer

   The one behind must be moving faster than the one in front. Only a faster particle behind can close the gap.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M2
• Pearson Edexcel IAL Mechanics M2 (WME02) question paper, June 2025: front-cover instructions on g and accuracy

Last updated 8 November 2026. Spotted a mistake? Tell us.