Two particles: relative motion

7 A particle \(P _ { 1 }\) is projected vertically upwards, from horizontal ground, with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle \(P _ { 2 }\) is projected vertically upwards from the top of a tower of height 25 m , with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the time for which \(P _ { 1 }\) is higher than the top of the tower,
2. the velocities of the particles at the instant when the particles are at the same height,
3. the time for which \(P _ { 1 }\) is higher than \(P _ { 2 }\) and is moving upwards.

5 Two particles \(P\) and \(Q\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(P\) and \(Q\) are \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively and the heights of \(P\) and \(Q\) above the ground, \(t\) seconds after projection, are \(h _ { P } \mathrm {~m}\) and \(h _ { Q } \mathrm {~m}\) respectively. Each particle comes to rest on returning to the ground.

1. Find the set of values of \(t\) for which the particles are travelling in opposite directions.
2. At a certain instant, \(P\) and \(Q\) are above the ground and \(3 h _ { P } = 8 h _ { Q }\). Find the velocities of \(P\) and \(Q\) at this instant.

5 A particle \(P\) is projected vertically upwards from a point on the ground with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) is projected vertically upwards from the same point with speed \(7 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is projected \(T\) seconds later than particle \(P\).

1. Given that the particles reach the ground at the same instant, find the value of \(T\).
2. At a certain instant when both \(P\) and \(Q\) are in motion, \(P\) is 5 m higher than \(Q\). Find the magnitude and direction of the velocity of each of the particles at this instant.

4 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find

1. the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height,
2. the height of \(A\) above the ground when \(B\) is 0.9 m above \(A\).

5 Particles \(P\) and \(Q\) are projected vertically upwards, from different points on horizontal ground, with velocities of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(Q\) is projected 0.4 s later than \(P\). Find

1. the time for which \(P\) 's height above the ground is greater than 15 m ,
2. the velocities of \(P\) and \(Q\) at the instant when the particles are at the same height.

5 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Write down expressions for the heights above the ground of \(A\) and \(B\) at time \(t\) seconds after projection.
2. Hence find a simplified expression for the difference in the heights of \(A\) and \(B\) at time \(t\) seconds after projection.
3. Find the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height. At the instant when \(B\) is 3.5 m above \(A\), find
4. whether \(A\) is moving upwards or downwards,
5. the height of \(A\) above the ground.