AS June 2025 Q2
2. A particle \(Q\) of mass \(3m\) is at rest on a smooth horizontal plane. A particle \(P\) of mass \(m\) is moving along the plane when it collides directly with \(Q\).
The speed of \(P\) immediately before the collision is \(u\).
The direction of motion of \(P\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
Given that \(e = \dfrac{1}{2}\)
| Scheme | Marks | AO |
|---|---|---|
| \[\begin{array}{ccc} u \rightarrow & \qquad & \rightarrow 0 \\ (P)\ m & & 3m\ (Q) \\ v \leftarrow & & \rightarrow w \end{array}\] | ||
| Use of CLM | M1 | 3.1a |
| \(-mv + 3mw = mu\) | A1 | 1.1b |
| Use of NEL | M1 | 3.4 |
| \(v + w = eu\) | A1 | 1.1b |
| Solve for \(v\): | M1 | 1.1b |
| \(\dfrac{u(3e-1)}{4}\)* | A1* | 2.2a |
| (6) |
Notes
N.B. When checking for consistency between their equations, mark the CLM equation FIRST.
M1: Use of CLM, with correct no. of terms, condone sign errors and consistent missing \(m\)’s
A1: Correct unsimplified equation. Allow \(v\) replaced by \(-v\)
M1: Use of NEL with \(e\) on the correct side of the equation, condone sign errors.
A1: Correct unsimplified equation consistent with CLM equation.
M1: Solve for \(v\) (must be dimensionally correct but allow slips in algebra)
A1*: Given answer correctly obtained, with no errors seen.
Allow \(\dfrac{u}{4}(3e-1)\) or \(\dfrac{1}{4}u(3e-1)\) or \(\dfrac{u}{4}(-1+3e)\) or \(\dfrac{1}{4}u(-1+3e)\) or \(\dfrac{u(-1+3e)}{4}\)
If they have \(v\) in the initial direction of \(P\) and obtain \(v = \dfrac{u(1-3e)}{4}\), we need to see a clear explanation of why the signs are changed.
| Scheme | Marks | AO |
|---|---|---|
| \(1 \geqslant e \gt \dfrac{1}{3}\) | B1 | 2.2a |
| (1) |
Notes
B1: cao
| Scheme | Marks | AO |
|---|---|---|
| Use of impulse-momentum for \(P\) or \(Q\) | M1 | 3.1a |
| \(P\): \(\pm m(v+u)\) OR \(Q\): \(\pm 3mw\) | A1 | 1.1b |
| \(\dfrac{9mu}{8}\) or \(1\dfrac{1}{8}mu\) | A1 | 1.1b |
| (3) | ||
| (10 marks) |
Notes
M1: Condone sign errors but must have correct terms (M0 if \(m\) omitted).
M0 if \(m\) is used with \(w\) or \(3m\) is used with \(v\).
A1: \(\pm m(v+u)\) or \(\pm 3mw\).
N.B. \(v\) and \(w\) do not need to be substituted.
A1: Accept \(1.1mu\) or better.
N.B. Must be of form \(kmu\) and must be positive.