Collision on inclined plane

7
\includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-12_399_1121_262_511} A particle \(P\) of mass 0.3 kg , lying on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\).

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the coefficient of friction between \(P\) and the horizontal plane.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are \(3 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) respectively. In the collision between the particles, the speed of \(A\) is reduced to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(B\) immediately after the collision.
   After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by \(90 \%\). The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.
2. Show that the speed of \(B\) immediately after it hits the barrier is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5
\includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_362_1065_258_539} Three particles \(P , Q\) and \(R\) lie on a line of greatest slope of a smooth inclined plane. \(P\) has mass 0.5 kg and initially is at the foot of the plane. \(R\) has mass 0.3 kg and initially is at the top of the plane. \(Q\) has mass 0.2 kg and is between \(P\) and \(R\) (see diagram). \(P\) is projected up the line of greatest slope with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant when \(Q\) and \(R\) are released from rest. Each particle has an acceleration of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) down the plane.

1. \(P\) and \(Q\) collide 0.4 s after being set in motion. Immediately after the collision \(Q\) moves up the plane with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed and direction of motion of \(P\) immediately after the collision.
2. 0.6 s after its collision with \(P , Q\) collides with \(R\) and the two particles coalesce. Find the speed and direction of motion of the combined particle immediately after the collision

6 A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 Ns directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).
2. Given that the plane is smooth, find the speed of the particle at \(Q\). It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).
3. Given that the plane is inclined at \(30 ^ { \circ }\) to the horizontal, find the magnitude of the frictional force on the particle.