Projectile energy - finding speed or height

7. A particle \(P\) is projected from a fixed point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. When \(P\) passes through the point \(B\) on its path, it has speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the vertical distance between \(A\) and \(B\). The minimum speed of \(P\) on its path from \(A\) to \(B\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the size of angle \(\alpha\).
3. Find the horizontal distance between \(A\) and \(B\).

5. \begin{figure}[h] \end{figure} A smooth straight ramp is fixed to horizontal ground. The ramp has length 8 m and is inclined at \(30 ^ { \circ }\) to the ground, as shown in Figure 2. A particle \(P\) of mass 0.7 kg is projected from a point \(A\) at the bottom of the ramp, up a line of greatest slope of the ramp, with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As \(P\) reaches the point \(B\) at the top of the ramp, \(P\) has speed \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the value of \(u\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the ground at a point \(C\). Immediately before hitting the ground at \(C\), particle \(P\) is moving at \(\theta ^ { \circ }\) below the horizontal with speed \(w \mathrm {~ms} ^ { - 1 }\). Find
2. 1. the value of \(w\),
   2. the value of \(\theta\),
3. the horizontal distance from \(B\) to \(C\).

6. A particle \(P\) is projected from a fixed point \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. As \(P\) passes through the point \(B\) on its path, \(P\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\beta\) below the horizontal.

1. By considering energy, find the vertical distance between \(A\) and \(B\). Particle \(P\) takes 1.5 seconds to travel from \(A\) to \(B\).
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the length of time for which the speed of \(P\) is less than \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.

1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
4. Find the horizontal distance from \(A\) to \(B\).

8. \begin{figure}[h] \end{figure} The fixed point \(A\) is \(h\) metres vertically above the point \(O\) that is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves freely under gravity. At time \(t = 2.5\) seconds, \(P\) strikes the ground at the point \(B\). At the instant when \(P\) strikes the ground, the speed of \(P\) is \(18 \mathrm {~ms} ^ { - 1 }\), as shown in Figure 4.

1. By considering energy, find the value of \(h\).
2. Find the distance \(O B\). As \(P\) moves from \(A\) to \(B\), the speed of \(P\) is less than or equal to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds.
3. Find the value of \(T\)

6. \begin{figure}[h] \end{figure} A ball is projected from a point \(A\) which is 8 m above horizontal ground as shown in Figure 4. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\). The speed of the ball immediately before it hits the ground is \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the value of \(u\). The time taken for the ball to move from \(A\) to \(B\) is 2 seconds. Find
2. the value of \(\theta\),
3. the minimum speed of the ball on its path from \(A\) to \(B\).

A stone, of mass 1.5 kg, is projected horizontally with speed 4 ms\(^{-1}\) from a height of 7 m above horizontal ground.

1. Show that the stone travels about 4.78 m horizontally before it hits the ground. [4 marks]
2. Find the height of the stone above the ground when it has travelled half of this horizontal distance. [4 marks]
3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground. [2 marks]
4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground. [3 marks]
5. State two modelling assumptions that you have made in answering this question. [2 marks]