M4 June 2016 Q2

EdexcelOld spec9 marksCollisions: 1 Sphere Oblique

Figure 2: ball path from A to X on the first wall at angle alpha, then to Y on the perpendicular wall through C, then rebounding parallel to XA — Figure 2

A small spherical ball \(P\) is at rest at the point \(A\) on a smooth horizontal floor. The ball is struck and travels along the floor until it hits a fixed smooth vertical wall at the point \(X\). The angle between \(AX\) and this wall is \(\alpha\), where \(\alpha\) is acute. A second fixed smooth vertical wall is perpendicular to the first wall and meets it in a vertical line through the point \(C\) on the floor. The ball rebounds from the first wall and hits the second wall at the point \(Y\). After \(P\) rebounds from the second wall, \(P\) is travelling in a direction parallel to \(XA\), as shown in Figure 2. The coefficient of restitution between the ball and the first wall is \(e\). The coefficient of restitution between the ball and the second wall is \(ke\).

Find the value of \(k\). (9)

Diagram: path with speed U from A to X at angle alpha to the first wall, then to Y, then leaving Y at angle 90 minus alpha to the second wall — Mark scheme

Velocity components: first collision U cos alpha parallel and e U sin alpha perpendicular to the first wall; second collision k e U cos alpha and e U sin alpha — Mark scheme

Scheme	Marks
First impact:
Component parallel to wall: \(= U\cos\alpha\)	B1
Perp to wall: NLR: \(eU\sin\alpha\)	M1
	A1
Second impact:
parallel to wall vel after \(= eU\sin\alpha\)	B1
Perp to wall \(ke\times U\cos\alpha\)	B1
Direction at \((90 - \alpha)\) to the wall	B1
\(\Rightarrow \tan(90 - \alpha) = \dfrac{keU\cos\alpha}{Ue\sin\alpha}\) or \(\tan\alpha = \dfrac{eU\sin\alpha}{keU\cos\alpha}\)	M1
\(\cot\alpha = k\cot\alpha\) or \(\tan\alpha = \dfrac{1}{k}\tan\alpha\)	A1
\(k = 1\)	A1
	(9 marks)

Notes