Vertical motion under gravity

A particle \(P\) is projected vertically upwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) which is \(2.8\text{ m}\) above horizontal ground.

1. Find the greatest height above the ground reached by \(P\). [3]
2. Find the length of time for which \(P\) is at a height of more than \(3.6\text{ m}\) above the ground. [4]

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]

1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]

1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]

The points \(P\) and \(Q\) are at the same height \(h\) metres above horizontal ground. A small stone is dropped from rest from \(P\). Half a second later a second small stone is thrown vertically downwards from \(Q\) with speed 7.35 m s\(^{-1}\). Given that the stones hit the ground at the same time, find the value of \(h\). [7]

A small stone is projected vertically upwards from the point \(O\) and moves freely under gravity. The point \(A\) is 3.6 m vertically above \(O\). When the stone first reaches \(A\), the stone is moving upwards with speed 11.2 m s\(^{-1}\). The stone is modelled as a particle.

1. Find the maximum height above \(O\) reached by the stone. [4]
2. Find the total time between the instant when the stone was projected from \(O\) and the instant when it returns to \(O\). [5]
3. Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through \(A\) moving upwards to the instant when it returns to \(O\). Show, on the axes, the coordinates of the points where your graph meets the axes. [4]

A stone is released from rest on a bridge and falls vertically into a lake. The stone has velocity \(14\text{ m s}^{-1}\) when it enters the lake.

1. Calculate the distance the stone falls before it enters the lake, and the time after its release when it enters the lake. [4]

The lake is \(15\text{ m}\) deep and the stone has velocity \(20\text{ m s}^{-1}\) immediately before it reaches the bed of the lake.

1. Given that there is no sudden change in the velocity of the stone when it enters the lake, find the acceleration of the stone while it is falling through the lake. [3]

A small object, of mass 0.02 kg, is dropped from rest from the top of a building which is160 m high.

1. Calculate the speed of the object as it hits the ground. [3]
2. Determine the time taken for the object to reach the ground. [3]
3. State one assumption you have made in your solution. [1]