Vertical projection: time to ground

3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find

1. her speed when she reaches the surface of the pool,
2. the time taken to reach the surface of the pool.
3. State two physical factors which have been ignored in the model.

1. A small stone is projected vertically upwards with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

1. the speed of the stone at the instant when it is 2 m above the ground,
2. the total time between the instant when the stone is projected from \(O\) and the instant when it first strikes the ground.

4. At time \(t = 0\), a small ball is projected vertically upwards from a point \(A\) which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point \(B\), where \(A B = 19.6 \mathrm {~m}\) and first hits the ground at time \(t = T\) seconds. The ball is modelled as a particle moving freely under gravity.

1. Find the value of \(T\).
2. Sketch a speed-time graph for the motion of the ball from \(t = 0\) to \(t = T\) seconds.
   (No further calculations are needed in order to draw this sketch.)

1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.

1. Find the value of \(V\).
2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
3. Find the length of time for which the particle is above \(C\).
4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)

2. A ball is thrown vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is \(h\) metres above the ground. The ball moves freely under gravity until it hits the ground 5 s later.

1. Find the value of \(h\). A second ball is thrown vertically downwards with speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and moves freely under gravity until it hits the ground. The first ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the second ball hits the ground with speed \(\frac { 3 } { 4 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(w\).

2. A small ball is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point that is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity. Find

1. the total time from when the ball is thrown to when it first hits the ground,
2. the speed of the ball immediately before it first hits the ground,
3. the total distance travelled by the ball from when it is thrown to when it first hits the ground.
4. Sketch a velocity-time graph for the motion of the ball from when it is thrown to when it first hits the ground. State the coordinates of the start point and the coordinates of the end point of your graph.
   DO NOT WRITEIN THIS AREA

6. \begin{figure}[h] \end{figure} A small ball is thrown vertically upwards at time \(t = 0\) from a point \(A\) which is above horizontal ground. The ball hits the ground 7 s later. The ball is modelled as a particle moving freely under gravity.
The velocity-time graph shown in Figure 3 represents the motion of the ball for \(0 \leqslant t \leqslant 7\)

1. Find the speed with which the ball is thrown.
2. Find the height of \(A\) above the ground.

1 A particle \(P\) is projected vertically downwards from a fixed point \(O\) with initial speed \(4.2 \mathrm {~ms} ^ { - 1 }\), and takes 1.5 s to reach the ground. Calculate

1. the speed of \(P\) when it reaches the ground,
2. the height of \(O\) above the ground,
3. the speed of \(P\) when it is 5 m above the ground.

10 A small ball \(B\) is projected vertically upwards from a point 2 m above horizontal ground. \(B\) is projected with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\), and takes \(t\) seconds to reach the ground. Find the value of \(t\).

1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the initial kinetic energy of the box.
2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
3. Hence find the speed of the box when it hits the ground.
4. State two modelling assumptions which you have made.

1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).