M4 June 2014 Q3

EdexcelOld spec8 marksCollisions: 1 Sphere Oblique

3. A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\dfrac{1}{3}\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision.

Find the angle through which the path of the ball is deflected by the collision. (8)

alt1 diagram: velocity components y perpendicular and x parallel to the wall before; y/3 and x after — Mark scheme

alt2 diagram: components u sin theta and u cos theta before; u sin theta over 3 and u cos theta after; angles theta and alpha with the wall — Mark scheme

Scheme	Marks
alt1
Speed perpendicular to wall after collision \(= \dfrac{y}{3}\)	B1
Speed parallel to the wall is unchanged	B1
\(\dfrac{1}{4}\left(x^2 + y^2\right) = x^2 + \dfrac{1}{9}y^2\)	M1 A1
\(9\left(x^2 + y^2\right) = 4\left(9x^2 + y^2\right),\ \ 27x^2 = 5y^2,\ \ x = \sqrt{\dfrac{5}{27}}\,y\)	A1
direction deflected by \(\tan^{-1}\dfrac{y}{x} + \tan^{-1}\dfrac{y}{3x}\)	M1 A1
\(= \tan^{-1}\sqrt{\dfrac{27}{5}} + \tan^{-1}\sqrt{\dfrac{3}{5}} = 104.5^\circ \quad (104)\)	A1
	(8)
	(8 marks)

Speed perpendicular to wall after collision \(= \dfrac{u\sin\theta}{3}\)	B1
Speed parallel to the wall is unchanged	B1
\(\dfrac{u^2}{4} = \dfrac{u^2}{9}\sin^2\theta + u^2\cos^2\theta\)	M1 A1
\(27\cos^2\theta = 5\sin^2\theta,\ \ \tan^2\theta = \dfrac{27}{5}\)	A1
deflected by \(\theta + \alpha\), \(\tan(\theta + \alpha) = \dfrac{\tan\theta + \frac{1}{3}\tan\theta}{1 - \frac{1}{3}\tan^2\theta} \quad \left(= -\sqrt{15}\right)\)	M1 A1
\(\theta + \alpha = 104.5^\circ \quad (104)\)	A1

Notes

alt2