Finding angle given constraints

6 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(\theta\).
2. Show that at the instant 4 s after projection the particle is 33.75 m below the level of the point of projection and find the direction of motion at this instant.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-12_259_609_255_769} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows an object made from a uniform wire of length 0.8 m . The object consists of a straight part \(A B\), and a semicircular part \(B C\) such that \(A , B\) and \(C\) lie in the same straight line. The radius of the semicircle is \(r \mathrm {~m}\) and the centre of mass of the object is 0.1 m from line \(A B C\).
3. Show that \(r = 0.2\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-13_615_383_260_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The object is freely suspended at \(A\) and a horizontal force of magnitude 7 N is applied to the object at \(C\) so that the object is in equilibrium with \(A B C\) vertical (see Fig. 2).
4. Calculate the weight of the object.
   The 7 N force is removed and the object hangs in equilibrium with \(A B C\) at an angle of \(\theta ^ { \circ }\) with the vertical.
5. Find \(\theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 A particle is projected from a point on horizontal ground. At the instant 2 s after projection, the particle has travelled a horizontal distance of 30 m and is at its greatest height above the ground. Find the initial speed and the angle of projection of the particle.

1 A golf ball \(B\) is projected from a point \(O\) on horizontal ground. \(B\) hits the ground for the first time at a point 48 m away from \(O\) at time 2.4 s after projection. Calculate the angle of projection.

2 A small ball is projected from a point 1.5 m above horizontal ground. At a point 9 m above the ground the ball is travelling at \(45 ^ { \circ }\) above the horizontal and its velocity is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle of projection of the ball.

4 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
2. Show that \(\cos \alpha = 0.28\).
3. Calculate the greatest height reached by the particle.

1 A stone is projected from a point on level ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(\theta ^ { \circ }\) above the horizontal. When the stone is at its greatest height it just passes over the top of a tree that is 17 m high. Calculate \(\theta\).

1 A football is kicked from horizontal ground with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The greatest height the football reaches above ground level is 2.44 m . By modelling the football as a particle and ignoring air resistance, find

1. the value of \(\theta\),
2. the range of the football.

\includegraphics{figure_2} A particle \(P\) is projected from a point \(O\) at an angle of \(60°\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45°\) (see diagram).

1. Show that the speed of projection of \(P\) is 8.20 m s\(^{-1}\), correct to 3 significant figures. [4]
2. Calculate the time after projection when the direction of motion of \(P\) is \(45°\) above the horizontal. [3]

A particle \(P\) is projected with speed \(V\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on horizontal ground. At the instant \(2\) s after projection, \(OP\) makes an angle of \(15°\) above the horizontal. Calculate \(V\). [4]

At time \(t\)s, a particle \(P\) is projected with speed \(40\)m s\(^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H\)m and the corresponding time is \(T\)s.

1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\). [2]

During the time between \(t = T\) and \(t = 3\), \(P\) descends a distance \(\frac{1}{4}H\).

1. Find the value of \(\theta\). [4]
2. Find the speed of \(P\) when \(t = 3\). [3]