Collision with unchanged direction

1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(k m\). The particles are moving towards each other on the same straight line on a smooth horizontal surface.
   The particles collide directly.
   Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(v\).

The direction of motion of \(P\) is unchanged by the collision.

1. Show that \(v = \frac { ( 3 - 3 k ) } { k } u\)
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that \(v \neq u\)
3. find the range of possible values of \(k\).

2. Two small smooth spheres \(P\) and \(Q\) are moving along a straight line in opposite directions, with equal speeds, and collide directly. Immediately after the impact, the direction of \(P\) 's motion has been reversed and its speed has been halved. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Express the speed of \(Q\) after the impact in the form \(a u ( b e + c )\), where \(a , b\) and \(c\) are constants to be found.
2. Deduce the range of values of \(e\) for which the direction of motion of \(Q\) remains unaltered.

7. Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9 m\) and \(4 m\) respectively, are moving towards each other along a straight line with speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(6 \mathrm {~ms} ^ { - 1 }\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.

1. Show that the speed of \(B\) after the impact cannot be more than \(3 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
2. Show that \(e < \frac { 3 } { 10 }\).
3. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\).

6. A smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2 m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,

1. show that the speed of \(Q\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + e ) u\),
2. find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 4 }\),
3. find the loss of kinetic energy in the collision.
4. Give one possible form of energy into which the lost kinetic energy has been transformed.

7 Two smooth, equally sized spheres, \(A\) and \(B\), are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, \(A\) has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the spheres is \(e\) 7

1. 1. Show that \(A\) does not change its direction of motion as a result of the collision.
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2. (ii) Find the value of \(e\)
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3. Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)

8 Two spheres, \(A\) and \(B\), of equal size are moving in the same direction along a straight line on a smooth horizontal surface. Sphere \(A\) has mass \(m\) and is moving with speed \(4 u\) Sphere \(B\) has mass \(6 m\) and is moving with speed \(u\)
The diagram shows the spheres and their velocities. \begin{figure}[h] \end{figure} B Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\) 8

1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision.
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2. Show that the speed of \(B\) immediately after the collision is \(\frac { u ( 3 e + 10 ) } { 7 }\)
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3. After the collision sphere \(A\) moves in the opposite direction.
   Find the range of possible values for \(e\)
   [0pt] [5 marks]