A2 June 2023 Q3
3. A particle \(P\) of mass \(2m\) is moving in a straight line with speed \(3u\) on a smooth horizontal plane. It collides directly with a particle \(Q\) of mass \(m\) that is moving on the plane with speed \(2u\) in the opposite direction to \(P\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \dfrac{4}{5}\)
After the collision \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
| Scheme | Marks | AO |
|---|---|---|
If it helps the candidate, ignore their diagram. | ||
| CLM: | M1 | 3.1a |
| \(2m \times 3u - m \times 2u = 2mv_P + mv_Q \qquad (4u = 2v_P + v_Q)\) | A1 | 1.1b |
| Impact Law: | M1 | 3.4 |
| \(5ue = -v_P + v_Q\) | A1 | 1.1b |
| Attempts to solve for \(v_Q\) | dM1 | 2.1 |
| \(v_Q = \dfrac{(4 + 10e)u}{3}\) * | A1* | 2.2a |
| (6) |
Notes
M1: CLM used. Dimensionally correct, mass \(\times\) velocity. All terms required. Condone sign errors. Condone consistent \(g\)’s or cancelled \(m\)’s.
A1: Correct unsimplified equation
M1: NEL used correctly with \(e\) appearing on the correct side of the equation. Condone sign errors, must have the correct number of terms.
A1: Correct unsimplified equation. Direction of \(v_Q\) and \(v_P\) must be consistent with their CLM equation.
dM1: Use their correctly formed equations to solve for \(v_Q\) At least one line of working should be seen.
A1*: Correct given answer correctly obtained \(v_Q = \dfrac{(4 + 10e)u}{3}\) *
Also accept: \(\dfrac{1}{3}(4 + 10e)u \qquad \dfrac{u(4 + 10e)}{3} \qquad \dfrac{u}{3}(4 + 10e)\)
Do not accept the 4 and \(10e\) reversed eg \(\dfrac{(10e + 4)u}{3}\) is A0*
| Scheme | Marks | AO |
|---|---|---|
| \(v_P = \dfrac{(4 - 5e)u}{3}\) oe | M1 | 1.1b |
| Correct rebound speed or velocity of \(Q\) seen \(\pm\dfrac{f(4 + 10e)u}{3}\) | B1 | 3.4 |
| States a correct inequality Eg 2nd collision if
| M1 | 3.1a |
| \((1 \geqslant)\, f > \dfrac{5e - 4}{4 + 10e}\) | A1 | 1.1b |
| (4) | ||
| (10 marks) |
Notes
M1: Attempt to solve for \(v_P\). If \(v_P\) is found in (a) it must be used in (b) to score this mark.
Note that if \(P\) is assumed to reverse direction in (a) then \(v_P = \dfrac{(5e - 4)u}{3}\) oe
B1: Correct expression seen for speed or velocity of \(Q\) after rebound \(\pm\dfrac{f(4 + 10e)u}{3}\). This may appear on a diagram.
M1: Correct unsimplified inequality seen. The inequality must be correct, accepts cancelled \(u\)’s and/or 3’s
A1: Correct inequality, do not ISW.
Allow \(1 \geqslant f\) to be omitted but do not allow the strict inequality \(1 > f\).