M4 June 2005 Q1

EdexcelOld spec7 marksCollisions: 1 Sphere Oblique

1. A small smooth ball of mass \(\tfrac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \tfrac{3}{4}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\tfrac{1}{2}\).

Find

(a) the speed, to 3 significant figures, of the ball immediately after the impact, (5)

(b) the magnitude of the impulse received by the ball as it strikes the plane. (2)

Ball striking plane inclined at alpha: before, components u cos alpha into the plane and u sin alpha down the plane; after, v perpendicular to the plane and u sin alpha along it; impulse I normal to the plane — Mark scheme (a)

Scheme	Marks
Cpt along plane after impact \(= 10\sin\alpha = 10 \times \tfrac{3}{5} = 6\)	B1
\(v = e \times 10\cos\alpha\ \ \left(= \tfrac{1}{2} \times 10 \times \tfrac{4}{5} = 4\right)\)	M1 A1
Speed \(= \sqrt{4^2 + 6^2} = 7.21\) ms\(^{-1}\) (3SF)	M1 A1
	(5)

Scheme	Marks
\(I = \tfrac{1}{2}(4 - {-8}) = 6\) Ns	M1 A1
	(2)
	(7 marks)

Notes