Collision sequences & checks | CAIE 9709 Mechanics Notes

Mechanics

Mathematics9709

Treat each collision as a separate event

Write the order of bodies on the line.
Complete the first momentum equation.
Carry every resulting velocity into the next snapshot.
Write a new momentum equation for the next collision only.
Keep the same positive direction throughout the sequence.

Worked example

A two-collision chain

Particles A, B and C have masses 1 kg, 2 kg and 3 kg. B and C are initially at rest. A moves right at 6 m s⁻¹, hits B and comes to rest. B then hits C and comes to rest.

First collision:

1(6)+2(0)=1(0)+2v_B

v_B=3\text{ m s}^{-1}

Second collision:

2(3)+3(0)=2(0)+3v_C

v_C=2\text{ m s}^{-1}

Examiner note

A 2024 question required two successive coalescing collisions. The mark scheme awarded separate method marks for the two conservation equations and allowed the first result to be followed through into the second. (9709/42/M/J/24 Q6(a))

Check catch-up and no-further-impact conditions

Swipe left or right to see the whole diagram →

The algebra must match the story. Check catch-up before a collision and relative ordering after the last collision.

If two bodies move in the same direction, the rear body must be faster to catch the front body.
If they move towards one another, the distance between them must be decreasing. With right positive, this means the body on the left has the greater signed velocity.
After the final collision, compare positions and velocities to see whether one body can catch another again.
An algebraic answer can still be rejected if it contradicts the collision order.

Common mistake

“No further impact” is not a phrase to copy into the conclusion. Check how the distance between the bodies changes, using their order, directions and speeds.

Finish with a five-part reasonableness check

Were all masses expressed in the same unit?
Did every velocity sign match one chosen positive direction?
Was every body in the chosen system included before and after?
Does a negative answer have a stated physical direction?
Does the collision order remain physically possible?

Key idea

The signed conservation equation is the method and must be shown. If its rearrangement is immediate, write it directly. If the arithmetic is awkward, using the calculator's equation solver is a fast main route. Record the resulting velocity and then interpret its sign.

The current syllabus does not require impulse or coefficient of restitution. Do not add either unless a future syllabus explicitly introduces it.

Your turnIndependent check [6 marks]

Particles A, B and C of masses 0.2 kg, 0.3 kg and 0.5 kg lie in that order on a smooth horizontal line. B and C are initially at rest. A moves right at 5 m s⁻¹ and hits B. After the first impact, A rebounds left at 1 m s⁻¹. B then hits C and comes to rest. Find B's velocity after the first impact, C's velocity after the second impact, and explain why the stated second collision is possible.

Show worked answer

Take right as positive.

p_{\text{before}}=0.2(5)=1

p_{\text{after}}=0.2(-1)+0.3v_B

1=-0.2+0.3v_B

v_B=4\text{ m s}^{-1}

p_{\text{before}}=0.3(4)=1.2

p_{\text{after}}=0.5v_C

v_C=2.4\text{ m s}^{-1}

B is behind C and moves right at 4 m s⁻¹ while C is at rest, so B catches C. The second collision is possible.

Answer

v_B=+4\text{ m s}^{-1}

v_C=+2.4\text{ m s}^{-1}