Two projectiles meeting - vertical motion only

A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \text{ ms}^{-1}\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \text{ ms}^{-1}\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).

1. Show that \(T = 3.5\). [4]
2. Find the height above \(O\) at which the particles collide. [1]
3. Find the time from \(A\) being projected until \(C\) returns to \(O\). [5]

A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \text{ m s}^{-1}\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet. [5]

Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). One second later particle \(B\) is dropped from rest from a height of 40 m.

1. Find the height above the ground at which the two particles collide. [4]
2. Find the difference in the speeds of the two particles at the instant when the collision occurs. [3]

Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H\) m directly above A. \includegraphics{figure_5} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4\text{ms}^{-1}\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V\text{ms}^{-1}\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). [7]