Range of coefficient of restitution

1 Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2 m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4 u\) and \(3 u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision.

1. A particle \(P\) of mass \(2 m\) and a particle \(Q\) of mass \(5 m\) are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The direction of motion of \(Q\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 3 }\)
2. show that the kinetic energy lost in the collision is \(\frac { 40 m u ^ { 2 } } { 7 }\).
3. Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if \(e > \frac { 1 } { 3 }\)

7 Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3 m\) and \(m\) respectively.
\(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest.
\(P\) subsequently collides with \(Q\).
\includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-10_273_864_685_589} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\).
The coefficient of restitution between the spheres is \(e\).
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1. 1. Show that $$v = \frac { u ( 3 - e ) } { 4 }$$ 7
2. (ii) Find \(w\), in terms of \(e\) and \(u\), simplifying your answer.
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3. Deduce that $$\frac { u } { 2 } \leq v \leq \frac { 3 u } { 4 }$$ 7
4. 1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\).
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5. (ii) Describe the impulse that \(Q\) exerts on \(P\).
   \includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-13_2488_1719_219_150} Question number Additional page, if required.
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