A2 June 2022 Q4
4.
Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3m\) and the mass of \(B\) is \(4m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3u\) at \(30^\circ\) to the line of centres of the spheres and \(B\) is moving with speed \(2u\) at \(30^\circ\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90^\circ\) by the collision, as shown in Figure 3.
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Parallel to line of centres: | M1 | 3.1b |
| \(9mu\cos 30^\circ - 8mu\cos 30^\circ = 4mv\cos 60^\circ - 3mw\cos\theta\) \((u\cos 30^\circ = 2v - 3w\cos\theta)\) \((u\cos 30^\circ = 2v - 3w_x)\) | A1 | 1.1b |
| \(\updownarrow A:\ (w_y =)\, w\sin\theta = 3u\sin 30^\circ \left(= \dfrac{3u}{2}\right)\) \(\updownarrow B:\ v\sin 60^\circ = 2u\sin 30^\circ\ (= u)\) \(\left(v = \dfrac{2u}{\sqrt{3}}\right)\) | B1 B1 | 3.4 3.4 |
| \((w_x =)\, w\cos\theta = \dfrac{1}{3}(2v - u\cos 30^\circ) = \dfrac{5u\sqrt{3}}{18}\) \((w_y =)\, w\sin\theta = \dfrac{3u}{2}\) \(\left(\Rightarrow \tan\theta = \dfrac{9\sqrt{3}}{5},\ \ \theta = 72.2^\circ\right)\) | M1 | 1.1b |
| Direction deflected by \(77.8^\circ\) (\(78^\circ\) or better) | A1 | 2.2a |
Notes
M1: Use of CLM parallel to the line of centres.
Need all 4 terms. Dimensionally correct. Condone sign errors and sin/cos confusion.
A1: Correct unsimplified equation. Allow e.g. \(w_x\) in place of \(w\cos\theta\) and \(v_x\) in place of \(v\cos 60^\circ\).
Allow if they have divided through by a common factor e.g. \(m\)
NB there is no mark for the correct use of the impact law because the candidates are not required to find the coefficient of restitution. They might however find it as part of an alternative method. In this case, the M marks below are for a complete correct method to achieve the required result. Ignore work to find \(e\) if it is not used.
B1 B1: No change perpendicular to line of centres for one sphere. Allow e.g. \(w_y\) in place of \(w\sin\theta\).
Check the diagrams – the vertical components are often shown there.
No change perpendicular to line of centres for both spheres
M1: Use scalar product or solve simultaneous equations to find \(\theta\) for a relevant angle using their \(w_x\)
They need to get as far as \(\theta =\) a numerical value for a relevant angle
A1: \(78^\circ\) or better
| Scheme | Marks | AO |
|---|---|---|
| Magnitude of impulse | M1 | 3.1b |
| \(= 4m\big(v\cos 60^\circ - (-2u\cos 30^\circ)\big)\) | A1 | 1.1b |
| \(= 4m\left(\dfrac{1}{2} \times \dfrac{u}{\sin 60^\circ} - \left(-2u\dfrac{\sqrt{3}}{2}\right)\right) = \dfrac{16\sqrt{3}}{3}mu\) | A1 | 2.2a |
| OR: magnitude \(= 3m(3u\cos 30^\circ + w\cos\theta)\) \(= 3m\left(+\dfrac{5u\sqrt{3}}{18} - \left(-\dfrac{3u\sqrt{3}}{2}\right)\right) = \dfrac{16\sqrt{3}}{3}mu\) | ||
| (9) | ||
| (9 marks) |
Notes
M1: Use of \(I = mv - mu\) in direction of line of centres. Condone subtraction in either order
Allow M1 if they think that they have subtracted but they have not actually taken account of the change of direction.
Allow M1 if they go direct to the correct expression with a + without telling you that they have taken account of the change in direction
Allow M1 if they go straight to an unsimplified expression in surds using values already found earlier.
A1: Correct unsimplified expression.
Allow the negative of this
A1: Any equivalent simplified form. Must be positive. Condone if they change sign at the very end without explaining why. Accept \(9.2(376\ldots)mu\) (2 sf or better)
NB You might see candidates using the right angle and matrix multiplication to rotate the initial velocity of \(B\) to find the correct components of the velocity of \(B\) after impact.