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, 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.
      7
      1. (ii) Find the value of \(e\) 7
   2. Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)

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 \(2m\) 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 \(\frac{1}{3}(1 + e)u\), [5]

1. find the range of possible values of \(e\). [4]

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]

1. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

TURN OVER FOR QUESTION 7

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 \(2m\) 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 \(\frac{1}{3}(1 + e)u\), [5]
2. find the range of possible values of \(e\). [4]

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]
2. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.

1. Find the speeds of \(A\) and \(B\) immediately after the collision. [7]
2. Find the range of possible values of \(e\). [2]

After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).

1. Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\). [6]

A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision. Show that \(e > \frac{1}{3}\). [8]

Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 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 ms\(^{-1}\). [5 marks]

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(e < \frac{3}{10}\). [5 marks]
2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]

Two spheres A and B are free to move on a smooth horizontal surface. The masses of A and B are 2 kg and 3 kg respectively. Both A and B are initially at rest. Sphere A is set in motion directly towards sphere B with speed 4 m s\(^{-1}\) and subsequently collides with sphere B The coefficient of restitution between the spheres is \(e\)

1. 1. Show that the speed of B immediately after the collision is $$\frac{8(1 + e)}{5}$$ [4 marks]
   2. Find an expression, in terms of \(e\), for the velocity of A immediately after the collision. [2 marks]
2. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\) [2 marks]
3. 1. The impulse of sphere A on sphere B is \(I\) The impulse of sphere B on sphere A is \(J\) Given that the collision is perfectly inelastic, find the value of \(I + J\) [1 mark]
   2. State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic. [2 marks]

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 \(4u\) Sphere \(B\) has mass \(6m\) and is moving with speed \(u\) The diagram shows the spheres and their velocities. \includegraphics{figure_8} Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\)

1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision. [1 mark]
2. Show that the speed of \(B\) immediately after the collision is \(\frac{u(3e + 10)}{7}\) [4 marks]
3. After the collision sphere \(A\) moves in the opposite direction. Find the range of possible values for \(e\) [5 marks]