Vertical projection: speed at given height

7 A particle \(P\) of mass 0.2 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~ms} ^ { - 1 }\).

1. Show that the speed of \(P\) when it reaches 20 m above the ground is \(15 \mathrm {~ms} ^ { - 1 }\).
   When \(P\) reaches 20 m above the ground it collides with a second particle \(Q\) of mass 0.1 kg which is moving downwards at \(20 \mathrm {~ms} ^ { - 1 } . P\) is brought to instantaneous rest in the collision.
2. Find the velocity of \(Q\) immediately after the collision.
   When \(P\) reaches the ground it rebounds back directly upwards with half of the speed that it had immediately before hitting the ground.
3. Find the height above the ground at which \(P\) and \(Q\) next collide.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-3_476_1305_1519_420} A smooth slide \(A B\) is fixed so that its highest point \(A\) is 3 m above horizontal ground. \(B\) is \(h \mathrm {~m}\) above the ground. A particle \(P\) of mass 0.2 kg is released from rest at a point on the slide. The particle moves down the slide and, after passing \(B\), continues moving until it hits the ground (see diagram). The speed of \(P\) at \(B\) is \(v _ { B }\) and the speed at which \(P\) hits the ground is \(v _ { G }\).

1. In the case that \(P\) is released at \(A\), it is given that the kinetic energy of \(P\) at \(B\) is 1.6 J . Find
   1. the value of \(h\),
   2. the kinetic energy of the particle immediately before it reaches the ground,
   3. the ratio \(v _ { G } : v _ { B }\).
   4. In the case that \(P\) is released at the point \(X\) of the slide, which is \(H \mathrm {~m}\) above the ground (see diagram), it is given that \(v _ { G } : v _ { B } = 2.55\). Find the value of \(H\) correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-4_384_679_258_733} Particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(P\) hangs freely and \(Q\) is in contact with the table. A force of magnitude 3.2 N acts on \(Q\), upwards and away from the pulley, at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
      1. The system is in limiting equilibrium with \(P\) about to move upwards. Find the coefficient of friction between \(Q\) and the table. The force of magnitude 3.2 N is now removed and \(P\) starts to move downwards.
      2. Find the acceleration of the particles and the tension in the string.

3 A particle \(P\) is projected vertically upwards, from a point \(O\), with a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is the highest point reached by \(P\). Find

1. the speed of \(P\) when it is at the mid-point of \(O A\),
2. the time taken for \(P\) to reach the mid-point of \(O A\) while moving upwards.

4 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.