M4 June 2016 Q1

EdexcelOld spec8 marksCollisions: 2 Spheres Oblique

Figure 1: sphere A of mass m moving at angle alpha to the line of centres towards sphere B of mass 4m — Figure 1

A smooth uniform sphere \(A\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a second smooth uniform sphere \(B\), which is at rest on the plane. The sphere \(B\) has mass \(4m\) and the same radius as \(A\). Immediately before the collision the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres of the spheres, as shown in Figure 1. The direction of motion of \(A\) is turned through an angle of \(90^\circ\) by the collision and the coefficient of restitution between the spheres is \(\dfrac{1}{2}\)

Find the value of \(\tan\alpha\). (8)

Diagram: A (m) with speed u at angle alpha to the line of centres; after impact A has components v backwards along the line of centres and w perpendicular to it; B (4m) moves with speed x along the line of centres — Mark scheme

Diagram: A (m) with speed u at angle alpha to the line of centres; after impact A moves with speed v at angle 90 minus alpha to the line of centres; B (4m) moves with speed x — Mark scheme

Scheme	Marks
Along line of centres:
Con of mom: \(mu\cos\alpha = 4mx - mv\)	M1
\((u\cos\alpha = 4x - v)\)	A1
NLR: \(\dfrac{1}{2}u\cos\alpha = x + v\)	M1
\((2u\cos\alpha = 4x + 4v)\)	A1
\((5v = u\cos\alpha)\)
Perp to line of centres: no change to velocity so vel \(= w = u\sin\alpha\)	B1 (A1)
Deflected through \(90^\circ\) \(\left(\tan\alpha = \dfrac{v}{w}\right)\)	B1
\(\tan\alpha = \dfrac{\frac{1}{5}u\cos\alpha}{u\sin\alpha}\)	M1
\(\tan^2\alpha = \dfrac{1}{5}\qquad \tan\alpha = \sqrt{\dfrac{1}{5}}\) or 0.4472....	A1
	(8 marks)

Notes