M2 June 2016 Q7
7. Two particles \(A\) and \(B\), of mass \(2m\) and \(3m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\dfrac{3}{4}\)
After the collision \(B\) hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\dfrac{27}{4}mu\).
| Scheme | Marks |
|---|---|
| CLM: \(\ 6mu = 2mv + 3mw\) | M1 |
| \((6u = 2v + 3w)\) | A1 |
| Impact: \(\ w - v = \dfrac{3}{4} \times 3u\left(= \dfrac{9}{4}u\right)\) | M1 A1 |
| \(6u = 2w - \dfrac{9}{2}u + 3w\) | DM1 |
| \(w = \dfrac{21}{10}u = v_B\) | A1 |
| \(v = w - \dfrac{9}{4}u = \left(\dfrac{21}{10} - \dfrac{9}{4}\right)u = -\dfrac{3}{20}u,\ \ v_A = \dfrac{3}{20}u\) | A1 |
| (7) |
Notes
M1 Requires all 3 terms. Must be dimensionally correct. Condone sign error(s)
A1 This equation defines their directions
M1 Must be used with \(e\) on the correct side
A1 Penalise inconsistent directions here
DM1 Solve simultaneous equations for \(v\) or \(w\). Dependent on the 2 previous M marks
A1 One correct
A1 Both correct
| Scheme | Marks |
|---|---|
| Speed of \(B\) after hitting wall \(= \dfrac{21}{10}ue\) | M1(B1) |
| Impulse \(= \dfrac{27}{4}mu = 3m\left(\dfrac{21}{10}u + \dfrac{21}{10}ue\right)\) | M1 |
| \(\dfrac{9}{4} = \dfrac{21}{10}(1 + e),\ \ \ \ e = \dfrac{1}{14}\) | A1 |
| (3) |
Notes
M1(B1) \(e \times\) their \(w\)
M1 for their \(w\). Must be trying to use the correct equation with \(3m\).
7b alt
| Impulse \(= \dfrac{27}{4}mu = 3m\left(\dfrac{21}{10}u + V\right),\ \ \left(V = \dfrac{3u}{20}\right)\) | M1(B1) |
| \(\dfrac{21u}{10}e = \dfrac{3u}{20}\), | M1 |
| \(e = \dfrac{1}{14}\) | A1 (3) |
M1(B1) Use impulse to find \(V\). Must be trying to use the correct equation with \(3m\).
M1 \(V = e \times\) their \(w\).
| Scheme | Marks |
|---|---|
| Speed of \(B\) after second impact \(=\) \(\dfrac{1}{14} \times \dfrac{21}{10}u = \dfrac{3}{20}u\) | B1ft |
| Same velocity (and \(A\) has a head start), so no collision. | B1 |
| (2) | |
| (12 marks) |
Notes
B1ft Compare two relevant speeds. (ft on their \(V\) or their \(e\) x their \(w\))
B1 From correct work only