A2 October 2021 Q2
2. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
Particle \(A\) has mass \(5m\) and particle \(B\) has mass \(3m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e \gt 0\)
Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,
Given also that the kinetic energy of \(A\) immediately after the collision is 16% of the kinetic energy of \(A\) immediately before the collision,
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Use of CLM | M1 | 3.1a |
| \(5mv + 6mv\,(= 11mv) = 5mx - 3my\) \((11v = 5x - 3y)\) | A1 | 1.1b |
| Use of impact law | M1 | 3.1a |
| \(v = e(x + y)\) | A1 | 1.1b |
| \(\left\{\begin{aligned} 11ev &= 5ex - 3ey \\ 3v &= 3ex + 3ey \end{aligned}\right. \Rightarrow x = \dfrac{v}{8e}(11e + 3)\) | M1 | 3.1a |
| \(y = \dfrac{v}{8e}(5 - 11e)\) | A1 | 1.1b |
| \(e \gt 0\ \ (\Rightarrow\ x \gt 0)\ \Rightarrow\ 5 - 11e \gt 0\) | M1 | 3.4 |
| \(\Rightarrow 0 \lt e \lt \dfrac{5}{11}\) | A1 | 2.2a |
| (8) |
Notes
Correction: The second simultaneous equation is \(3v = 3ex + 3ey\) (corrected from the printed mark scheme: \(3v = 3ex - 3ey\)).
M1: All terms required. Dimensionally correct. Condone sign errors
A1: Correct unsimplified equation
M1: Used correctly. Condone sign errors
A1: Correct unsimplified equation
M1: Use their correctly formed equations to solve for \(v\) or \(w\) or a multiple of \(v\) or \(w\)
A1: Both velocities correct
M1: Use their velocities (in general form – not by considering one specific value) to form inequality for both moving in the same direction.
A1: Correct only.
| Scheme | Marks | AO |
|---|---|---|
| Form equation for KE | M1 | 2.1 |
| \(\dfrac{1}{2} \times 5m \times v^2 = \dfrac{16}{100} \times \dfrac{1}{2} \times 5m \times \dfrac{v^2}{64e^2}(11e + 3)^2\) | A1ft | 1.1b |
| \(\big(4(11e + 3) = (\pm)80e\big)\) \(e = \dfrac{1}{3}\) | A1 | 1.1b |
| Impulse \(= -5m(v - x)\) | M1 | 3.1a |
| \(= -5m\left(v - \dfrac{11v}{8} - \dfrac{3v}{8e}\right)\) Or: \(3m\left(2v + \dfrac{5v}{8e} - \dfrac{11v}{8}\right)\) | A1ft | 1.1b |
| Magnitude \(= \dfrac{15}{2}mv\) | A1 | 2.2a |
| (6) | ||
| (14 marks) |
Notes
M1: Dimensionally correct. Condone 16% on wrong side Allow \(M\) or \(5m\)
A1ft: Or equivalent. Correct unsimplified equation. Follow their \(x\) Allow \(M\) or \(5m\)
A1: Correct answer only Allow \(M\) or \(5m\)
M1: Correct use of \(I = mv - mu\). Must be subtracting.
A1ft: Accept \(\pm\) Follow their \(x\), \(y\), \(e\)
A1: Correct only. Must be positive.
Alternative (b)
| Scheme | Marks | AO |
|---|---|---|
| Form equation for KE | M1 | 2.1 |
| \(\dfrac{1}{2} \times 5m \times v^2 = \dfrac{16}{100} \times \dfrac{1}{2} \times 5m \times x^2\) | A1 | 1.1b |
| \(\Rightarrow x = \dfrac{5v}{2},\ y = \dfrac{v}{2}\ \ \Rightarrow e = \dfrac{1}{3}\) | A1 | 1.1b |
| Impulse \(= -5m(v - x)\) | M1 | 3.1a |
| \(= -5m\left(v - \dfrac{5v}{2}\right)\) Or: \(3m\left(2v + \dfrac{v}{2}\right)\) | A1 | 1,16 |
| Magnitude \(= \dfrac{15}{2}mv\) | A1 | 2.2a |
| (6) |