M4 June 2014 Q5

EdexcelOld spec12 marksCollisions: 2 Spheres Oblique Impulse & Momentum

Figure 1: spheres A(m) and B(3m) in contact; A moving with speed 3u at angle alpha to the line of centres, B moving with speed u at angle beta to the line of centres — Figure 1

Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\dfrac{1}{5}\). It is given that \(\cos\alpha = \dfrac{1}{3}\) and \(\cos\beta = \dfrac{2}{3}\) and that \(\alpha\) and \(\beta\) are both acute angles.

(a) Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\). (8)

(b) Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy. (4)

Before: A components u along and 3u sin alpha perpendicular; B components 2u/3 along and u sin beta perpendicular. After: A components x and 3u sin alpha; B components y and u sin beta — Mark scheme (a)

Scheme	Marks
CLM: \(mx + 3my = 3m \times u\cos\beta - m \times 3u\cos\alpha = mu \quad (x + 3y = u)\)	M1 A1
NEL: \(x - y = \dfrac{1}{5}(3u\cos\alpha + u\cos\beta)\left(= \dfrac{1}{5}\left(u + \dfrac{2}{3}u\right) = \dfrac{1}{3}u\right)\)	M1 A1
\(x = \dfrac{u}{2}\), or \(\ y = \dfrac{u}{6}\)	DM1 A1
Magnitude of the impulse on \(A\) \(= mu - \left(m \times -\dfrac{u}{2}\right) = \dfrac{3mu}{2}\)	M1 A1
	(8)

Mark scheme (b)

Scheme	Marks
Component of velocity perpendicular to the line of centres before = component after \(= 3u\sin\alpha = 3u \times \dfrac{\sqrt{8}}{3} = \sqrt{8}u\)	B1
KE lost \(= \dfrac{m}{2}\left(9u^2 - \left(8u^2 + \dfrac{1}{4}u^2\right)\right)\left[= \dfrac{3}{8}mu^2\right]\)	M1 A1
Fraction lost \(= \dfrac{3/8}{9/2} = \dfrac{3}{8} \times \dfrac{2}{9} = \dfrac{1}{12}\)	A1
	(4)
	(12 marks)

Notes

M1 Change in KE. Does not need to be a fraction at this stage. Does not need to include the (cancelling) component perpendicular to the line of centre.

A1 Correct unsimplified