A2 June 2022 Q5
5. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(P\) is \(3m\) and the mass of \(Q\) is \(4m\).
Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(u\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
After the collision with \(P\), particle \(Q\) collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(Q\).
The coefficient of restitution between \(Q\) and the wall is \(\dfrac{1}{2}\)
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Using CLM: | M1 | 3.4 |
| \(6mu - 4mu = -3mv + 4mw\) \((2u = -3v + 4w)\) | A1 | 1.1b |
| Use of impact law | M1 | 3.1a |
| \(w + v = e \times 3u\) | A1 | 1.1b |
| Complete method to find \(w\) | M1 | 2.1 |
| \(\left\{\begin{aligned} 3w + 3v &= 9eu \\ -3v + 4w &= 2u \end{aligned}\right. \Rightarrow 7w = 9eu + 2u,\quad w = \dfrac{u}{7}(9e + 2)\) * | A1* | 2.2a |
| (6) |
Notes
M1: Use of CLM. Need all terms. Must be dimensionally correct. Condone sign errors.
Accept consistent cancelling of \(m\)
A1: Correct unsimplified equation for CLM.
They can have \(v\) in either direction
M1: Correct use of the impact law (used the right way round)
Condone sign errors in finding speed of approach and speed of separation.
A1: Correct unsimplified equation. Signs consistent with equation for CLM.
M1: Complete method to find \(w\) e.g. by forming simultaneous equations using CLM and Impact Law and solving. This requires both of the preceding M marks
A1*: Obtain given answer from correct working.
Accept with \(2 + 9e\) in place of \(9e + 2\)
Check that the answer does follow from the working.
| Scheme | Marks | AO |
|---|---|---|
| \(w' = \dfrac{1}{2} \times \dfrac{u}{7}(9e + 2)\) \(\left(= \dfrac{u}{14}(9e + 2)\right)\) | B1 | 1.1b |
| \(v = \dfrac{u}{7}(12e - 2)\) | B1 | 1.1b |
| For a second collision: \(w' \gt v\) | M1 | 3.3 |
| \(9e + 2 \gt 2(12e - 2),\quad 0 \lt e \lt \dfrac{2}{5}\) | A1 | 1.1b |
| (4) | ||
| (10 marks) |
Notes
B1: Speed of \(Q\) after impact with the wall. Any equivalent form. Correct speed can be implied by a correct negative velocity.
B1: Speed of \(P\) after impact with \(Q\). Accept \(\pm\). Any equivalent form in \(u\) and \(e\) (seen or implied)
M1: Form correct inequality using their \(v\) and \(w'\). A correct inequality has \(P\) and \(Q\) both moving away from the wall
A1: Correct interval only. Accept unsimplified fraction. Need both ends of the interval. Must be strict inequality at both ends.