M2 June 2014 (R) Q7

EdexcelOld spec14 marksCollisions: 2 Spheres Direct

7. A particle \(P\) of mass \(2m\) is moving in a straight line with speed \(3u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

(a) Show that the speed of \(Q\) immediately after the collision is \(\dfrac{u}{5}(8e + 3)\) (6)

(b) Find the range of possible values of \(e\). (4)

The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(kT\). Given that \(e = \dfrac{1}{2}\)

Diagram: P (2m) moving at 3u towards Q (3m) moving at u; after impact P moves at x to the left and Q at y to the right — Mark scheme (a)

Scheme	Marks
\(6mu - 3mu = 3my - 2mx\)	M1
\(3y - 2x = 3u\)	A1
\(4ue = x + y\)	M1 A1
\(y = \dfrac{u}{5}(8e + 3)\ \ \ \ **\)	DM1 A1
	(6)

Mark scheme (b)

Scheme	Marks
\(x = 4ue - y\)	M1
\(x = \dfrac{1}{5}u(20e - 8e - 3) = \dfrac{3}{5}u(4e - 1)\)	A1
\(x > 0 \Rightarrow e > \dfrac{1}{4}\)	M1
\(\therefore \dfrac{1}{4} < e \leqslant 1\)	A1
	(4)

Notes

M1 Use one of their equations from (a) and \(\pm\) the given \(y\) to find \(x\).

A1 Any equivalent form. Accept \(\pm\)

M1 Use \(x > 0\) to solve find values of e. The inequality must match their \(x\).

A1 Need both limits.

Mark scheme (c)

Scheme	Marks
\(e = \dfrac{1}{2}\ \ \ x = \dfrac{3u}{5}\ \ \ \ y = \dfrac{u}{5}(4 + 3) = \dfrac{7u}{5}\)	B1
\(T = \dfrac{1}{2} \times 2m \times 9u^2 + \dfrac{1}{2} \times 3mu^2\ \ \left(= \dfrac{21mu^2}{2}\right)\)	M1
\(kT = \dfrac{1}{2} \times 2m \times \dfrac{9u^2}{25} + \dfrac{1}{2} \times 3m \times \dfrac{49u^2}{25}\ \ \left(= \dfrac{165mu^2}{50}\right)\)	M1
\(k = \dfrac{\frac{1}{2} \times 2 \times \frac{9}{25} + \frac{1}{2} \times 3 \times \frac{49}{25}}{\frac{1}{2} \times 2 \times 9 + \frac{1}{2} \times 3}\ \ \left(\text{or } \dfrac{165}{50} \div \dfrac{21}{2}\right) = \dfrac{11}{35}\)	A1
	(4)
	(14 marks)

Notes

B1 Allow \(x = -\dfrac{3u}{5}\)

M1 KE before or KE after, in terms of \(u\)

M1 Second KE in terms of \(u\) and use to find \(k\).

A1 Or equivalent. 0.314 or better