AS June 2023 Q1
1. Two particles, \(P\) and \(Q\), of masses \(3m\) and \(2m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.
Immediately before the collision, \(P\) is moving with speed \(2u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\dfrac{9mu}{2}\)
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
Given that the speed of \(Q\) immediately before the collision is \(u\),
| Scheme | Marks | AO |
|---|---|---|
| \[\begin{array}{cccc} & 2u \rightarrow & \leftarrow u & \\ \dfrac{9mu}{2} \longleftarrow & (P)\ 3m & (Q)\,2m & \longrightarrow \dfrac{9mu}{2} \\ & \rightarrow v & \rightarrow y & \end{array}\] | ||
| Use of Impulse-momentum principle for \(P\) | M1 | 3.1a |
| \(-\dfrac{9mu}{2} = 3m(v - 2u)\) | A1 | 1.1b |
| \(v = \dfrac{u}{2}\) | A1 | 1.1b |
| (3) |
Notes
M1: Use of Impulse-momentum principle for \(P\), condone sign errors but M0 if dimensionally incorrect e.g. if \(m\) missing or \(g\) included.
A1: Correct unsimplified equation (may have \(-v\))
A1: cao (must be positive)
| Scheme | Marks | AO |
|---|---|---|
| Use of Impulse-momentum principle for \(Q\) or CLM | M1 | 3.4 |
| \(\dfrac{9mu}{2} = 2m(y - -u)\) OR \(3m \times 2u - 2mu = 3m \times \dfrac{u}{2} + 2my\) \(\left(y = \dfrac{5u}{4}\right)\) | A1 | 1.1b |
| Newton’s Experimental Law | M1 | 3.4 |
| \(e = \dfrac{\dfrac{5u}{4} - \dfrac{u}{2}}{2u + u}\) | A1 | 1.1b |
| \(e = \dfrac{1}{4}\) oe | A1 | 1.1b |
| (5) | ||
| (8 marks) |
Notes
M1: Use of Impulse-momentum principle for \(Q\), must be using \(2m\), condone sign errors but M0 if dimensionally incorrect e.g. \(g\) included
Or use of CLM, condone sign errors and consistent omission of \(m\)’s or consistent extra \(g\)’s, with their \(v\). Must be using correct masses on all four terms.
A1: Correct unsimplified equation
M1: Use of NEL with their \(v\) and their \(y\), condone sign errors but M0 if ratio for \(e\) is inverted
A1: Correct unsimplified equation in \(u\) and \(e\) only
A1: cao