M2 June 2017 Q7
7. Two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, lie at rest on a smooth horizontal surface. Particle \(B\) lies between \(A\) and a smooth vertical wall which is perpendicular to the line joining \(A\) and \(B\). Particle \(B\) is projected with speed \(5u\) in a direction perpendicular to the wall and collides with the wall. The coefficient of restitution between \(B\) and the wall is \(\dfrac{3}{5}\).
After the collision with the wall, \(B\) rebounds from the wall and collides directly with \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
The kinetic energy of \(B\) immediately after it collides with \(A\) is one quarter of the kinetic energy of \(B\) immediately before it collides with \(A\).
| Scheme | Marks |
|---|---|
| Impact with wall: \(\ v = \dfrac{3}{5} \times 5u = 3u\) | B1 |
| Impulse \(\pm 4m\left(3u - (-5u)\right)\) | M1 |
| Magnitude \(32mu\) (Ns) | A1 |
| (3) |
Notes
B1 or \(-3u\)
M1 M0 if clearly using \(mv + mu\), otherwise bod
| Scheme | Marks |
|---|---|
| CLM: \(\ 3mx + 4mw = 4m \times 3u\) | M1 A1ft |
| Impact: \(\ x - w = e \times 3u\) | M1 A1ft |
| \(3m(w + 3eu) + 4mw = 7mw + 9emu = 12mu\) | |
| \(7w = u(12 - 9e)\) | DM1 |
| Use of \(e \leqslant 1\) in their \(w\): \(7w \geqslant 3u\) | M1 |
| Hence \(w > 0\) and \(A\) and \(B\) are moving in the same direction | A1 |
| (7) |
Notes
M1 Need all 4 terms. Condone sign errors. Use of 5u is M0
A1ft follow their 3u
M1 Used the right way round. Use of 5u is M0
A1ft follow their 3u signs consistent with CLM equation
DM1 Solve for \(w\) or \(kw\). Dependent on two preceding M marks
M1 Condone use of \(\lt\)
A1 Complete argument leading to *given answer*
| Scheme | Marks |
|---|---|
| KE of \(B\) before collision \(= \dfrac{1}{2} \times 4m \times (3u)^2\ \left(= 18mu^2\right)\) | B1 |
| \(\Rightarrow \dfrac{1}{2} \times 4m\left(\dfrac{u}{7}(12 - 9e)\right)^2 = \dfrac{1}{4}\left(\dfrac{1}{2} \times 4m \times 9u^2\right)\) | M1 |
| \(4(12 - 9e)^2 = 49 \times 9,\ \ (4 - 3e)^2 = \dfrac{49}{4}\) | A1 |
| \(e = \dfrac{1}{6}\) | A1 |
| (4) | |
| (14 marks) |
Notes
B1 follow their 3u. seen or implied
M1 Follow their \(w\). \(\dfrac{1}{4}\) on the right side.
A1 Correct equation in \(m\), \(u\) and \(e\)
7c alt
| KE of \(B\) before collision \(= \dfrac{1}{2} \times 4m \times (3u)^2\ \left(= 18mu^2\right)\) | B1 |
| \(\Rightarrow \dfrac{1}{2}4mw^2 = \dfrac{1}{4} \times \dfrac{1}{2} \times 4m(3u)^2\ \ \ \ \left(w = \dfrac{1}{2} \times 3u\right)\) | M1 |
| \(\dfrac{3}{7}(4 - 3e) = \dfrac{1}{2} \times 3\) | A1 |
| \(e = \dfrac{1}{6}\) | A1 |
B1 follow their 3u
M1 \(\dfrac{1}{4}\) on the right side.
A1 Correct equation in \(m\), \(u\) and \(e\) from correct work only
A1 0.17 or better from correct work only