A2 June 2025 Q1
1. A particle \(A\) has mass \(4m\) and a particle \(B\) has mass \(3m\). The particles are moving along the same straight line on a smooth horizontal table. The particles are moving in opposite directions towards each other when they collide directly.
As a result of the collision, the direction of motion of each particle is reversed.
Immediately before the collision, the speed of \(A\) is \(u\) and the speed of \(B\) is \(ku\), where \(k\) is a constant.
Immediately after the collision, the speed of \(A\) is \(2v\) and the speed of \(B\) is \(3v\).
The magnitude of the impulse received by \(A\) in the collision is \(20mv\).
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Impulse-momentum equation for \(A\) or other complete method to form an equation in \(u\) and \(v\) only (and \(m\)) | M1 | 3.4 |
| For \(A\): \(20mv = 4m\big(2v - (-u)\big)\) | A1 | 1.1b |
| \(u = 3v\) | A1 | 1.1b |
| (3) |
Notes
Working in parts (a) and (b) may be marked together.
M1: Use of \(I = m(v-u)\) on \(A\) or equivalent complete method to form an equation in terms of \(u\) and \(v\) (and \(m\)) only (not \(k\)). Must consider change in momenta but condone in the wrong order or velocity sign errors. Might see \((2v+u)\) rather than \((2v - -u)\). Dimensionally correct equation with correct mass and velocities pairings. If CLM equation is formed in (a), then an impulse-equation must also be used to eliminate \(k\) for this mark. Allow consistent omission of \(m\).
A1: Correct unsimplified equation in \(u\) and \(v\) (and \(m\))
A1: Correct only, ISW after \(u = 3v\)
A0 for only \(v = \dfrac{u}{3}\)
| Scheme | Marks | AO |
|---|---|---|
| ||
| Impulse momentum equation for \(B\) or use of CLM to form an equation in \(k\) (and \(u\), \(v\), \(m\)) | M1 | 3.4 |
| Either For \(B\): \(20mv = 3m\big(3v - (-ku)\big)\) Or CLM: \(4m(u) - 3m(ku) = 3m(3v) - 4m(2v)\) | A1ft | 1.1b |
| \(k = \dfrac{11}{9}\) | A1 | 1.1b |
| (3) | ||
| (6 marks) |
Notes
Working in parts (a) and (b) may be marked together.
M1: Correct use of impulse-momentum equation or CLM to form an equation in \(k\), \(u\), \(v\) (and \(m\)). All terms required, allow consistent omission of \(m\). Dimensionally correct.
A1ft: Correct unsimplified equation in \(k\), \(u\) and \(v\). No need to replace \(u\) and \(v\) (follow through their positive \(u\) if substituted)
A1: Correct exact value. Accept \(1\dfrac{2}{9}\) or recurring decimal \(1.\dot{2}\) with correct notation.