M4 June 2013 Q3
3.
Two smooth uniform spheres \(A\) and \(B\), of equal radius \(r\), have masses \(3m\) and \(2m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision they are moving with speeds \(u\) and \(2u\) respectively. The centres of the spheres are moving towards each other along parallel paths at a distance \(1.6r\) apart, as shown in Figure 2.
The coefficient of restitution between the two spheres is \(\dfrac{1}{6}\).
Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision. (10)
| Scheme | Marks |
|---|---|
| \(0.6u\) or \(u\cos\alpha\) | B1 |
| \(1.2u\) or \(2u\cos\alpha\) | B1 |
| \(2m \times 1.2u - 3m \times 0.6u = 3ma + 2mb\) | M1 |
| \((3a + 2b = 0.6u)\) | A1ft |
| \(e(1.2u + 0.6u) = a - b\) | M1 |
| \((a - b = 0.3u)\) | A1ft |
| DM1 | |
| \(a = 0.24u\) or \(b = -0.06u\) | A1 |
| \((1.2u - (-0.06u)) \times 2m = 2.52mu\) | M1 |
| or \((0.24u - (-0.6u)) \times 3m = 2.52mu\) | A1 |
| (10 marks) |
Notes
B1 component of the initial velocity of \(A\) parallel to the line of centres on impact
B1 component of the initial velocity of \(B\) parallel to the line of centres on impact
M1 CLM parallel to the line of centres. Requires all the terms.
A1ft Correct unsimplified for their \(0.6u\) and \(1.2u\)
M1 Restitution parallel to the line of centres. Must be used the right way round.
A1ft Correct unsimplified for their \(0.6u\) and \(1.2u\). If signs are inconsistent between the two equations, penalise here.
DM1 Solve a pair of simultaneous eqns in \(a\) & \(b\) for one of \(a\) & \(b\). Dependent on the two previous M marks.
A1 In terms of \(u\) only
M1 Find impulse on \(A\) or \(B\). Unsimplified. For their \(a\) or \(b\). Correct mass for the velocities used.
A1 \(\dfrac{63}{25}\)