Collision with friction after impact

3. A particle \(P\) of mass 2 kg is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass 4 kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).

1. Show that the speed of \(P\) immediately after the collision is \(\frac { 1 } { 5 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude 10 N . The distance between the point of collision and the point where \(P\) comes to rest is 1.6 m .
2. Calculate the value of \(u\).
   (5)

6. A railway truck \(P\) of mass 1500 kg is moving on a straight horizontal track. The truck \(P\) collides with a truck \(Q\) of 2500 kg at a point \(A\). Immediately before the collision, \(P\) and \(Q\) are moving in the same direction with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after the collision, the direction of motion of \(P\) is unchanged and its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the trucks as particles,

1. show that the speed of \(Q\) immediately after the collision is \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision at \(A\), the truck \(P\) is acted upon by a constant braking force of magnitude 500 N . The truck \(P\) comes to rest at the point \(B\).
2. Find the distance \(A B\). After the collision \(Q\) continues to move with constant speed \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance between \(P\) and \(Q\) at the instant when \(P\) comes to rest.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\). By modelling the balls as particles, find

1. the speed of \(B\) immediately after the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given. The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
3. Find the coefficient of friction between \(B\) and the table.

5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\).
By modelling the balls as particles, find

1. the speed of \(B\) immediately after the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given.
   (3 marks)
   The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
3. Find the coefficient of friction between \(B\) and the table.
   (6 marks)

4. In a physics experiment, two balls \(A\) and \(B\), of mass \(4 m\) and \(3 m\) respectively, are travelling towards one another on a straight horizontal track. Both balls are travelling with speed \(2 \mathrm {~ms} ^ { - 1 }\) immediately before they collide. As a result of the impact, \(A\) is brought to rest and the direction of motion of \(B\) is reversed.
Modelling the track as smooth and the balls as particles,

1. find the speed of \(B\) immediately after the collision. A student notices that after the collision, \(B\) comes to rest 0.2 m from \(A\).
2. Show that the coefficient of friction between \(B\) and the track is 0.113 , correct to 3 decimal places.