AS June 2022 Q4
4. A particle \(P\) of mass \(2m\) kg is moving with speed \(2u\ \text{m s}^{-1}\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3m\) kg which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v\ \text{m s}^{-1}\)
Given that the direction of motion of \(P\) is reversed by the collision,
After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds.
The coefficient of restitution between \(Q\) and the wall is \(\dfrac{1}{6}\)
Given that \(P\) and \(Q\) collide again,
| Scheme | Marks | AO |
|---|---|---|
| \[\begin{array}{cc} 2u \rightarrow & 0 \\ P\ (2m) & Q(3m) \\ w \leftarrow & \rightarrow v \end{array}\] | ||
| Use of CLM | M1 | 3.4 |
| \(2m \times 2u = -2mw + 3mv\) | A1 | 1.1b |
| Use of NEL | M1 | 3.4 |
| \(2ue = w + v\) | A1 | 1.1b |
| Solve for \(v\) | D M1 | 1.1b |
| \(v = \dfrac{4u(1+e)}{5}\)* | A1* | 2.2a |
| (6) |
Notes
M1: Correct no. of terms, condone sign errors, allow consistently cancelled \(m\)’s or extra \(g\)’s or common factors throughout
A1: Correct equation; they may have \(w\) instead of \(-w\)
M1: Correct no. of terms, condone sign errors. M0 if \(e\) on the wrong side of the equation
A1: Correct equation; they may have \(w\) instead of \(-w\)
DM1: Solve for \(v\), dependent on previous two marks
A1*: Correct answer correctly obtained
| Scheme | Marks | AO |
|---|---|---|
| Since \(0 \leqslant e \leqslant 1\), \(\dfrac{4u(1+0)}{5} \leqslant v \leqslant \dfrac{4u(1+1)}{5}\) | M1 | 3.1a |
| i.e. \(\dfrac{4u}{5} \leqslant v \leqslant \dfrac{8u}{5}\)* | A1* | 2.2a |
| (2) |
Notes
M1: Use of \(0 \leqslant e \leqslant 1\) in the given answer; allow use of \(e = 0\) and \(e = 1\) to obtain the min and max expressions
M1A0 for ‘verification’.
A1*: Correct answer correctly obtained (including use of max and min)
| Scheme | Marks | AO |
|---|---|---|
| Solve for \(w\) | M1 | 1.1b |
| \(w = \dfrac{2u(3e-2)}{5}\) oe \((\text{m s}^{-1})\) or \(\left|\dfrac{2u(2-3e)}{5}\right|\) oe | A1 | 1.1b |
| (2) |
Notes
M1: Solve for their \(w\)
A1: cao
| Scheme | Marks | AO |
|---|---|---|
| Speed of \(Q\) after hitting the wall \(= \dfrac{1}{6}v\ \ (\text{m s}^{-1})\) | M1 | 3.4 |
| For a further collision between \(P\) and \(Q\), \(\dfrac{1}{6}v \gt w\) | M1 | 3.1a |
| Substitute for \(v\) and \(w\) and solve for \(e\) | M1 | 1.1b |
| \(e \lt \dfrac{7}{8}\) | A1 | 1.1b |
| \(\dfrac{2}{3} \lt e \lt \dfrac{7}{8}\) | A1 | 1.1b |
| (5) | ||
| (15 marks) |
Notes
M1: Speed so must see a positive quantity
M0 if \(\dfrac{1}{6}\) is on the wrong side of the equation
M1: Correct inequality for their \(w\) (allow even if their \(w\) is dimensionally incorrect)
M1: Independent M mark but must have an inequality in \(v\) and \(w\):
Substitute for \(v\), using given answer, and \(w\) and solve for \(e\)
A1: Correct upper bound for \(e\)
A1: cao