1 Fig. 7 shows the trajectory of an object which is projected from a point O on horizontal ground. Its initial velocity is \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal. \begin{figure}[h] \end{figure}

1. Show that, according to the standard projectile model in which air resistance is neglected, the flight time, \(T \mathrm {~s}\), and the range, \(R \mathrm {~m}\), are given by $$T = \frac { 80 \sin \alpha } { g } \text { and } R = \frac { 3200 \sin \alpha \cos \alpha } { g }$$ A company is designing a new type of ball and wants to model its flight.
2. Initially the company uses the standard projectile model. Use this model to show that when \(\alpha = 30 ^ { \circ }\) and the initial speed is \(40 \mathrm {~ms} ^ { - 1 } , T\) is approximately 4.08 and \(R\) is approximately 141.4 . Find the values of \(T\) and \(R\) when \(\alpha = 45 ^ { \circ }\). The company tests the ball using a machine that projects it from ground level across horizontal ground. The speed of projection is set at \(40 \mathrm {~ms} ^ { - 1 }\). When the angle of projection is set at \(30 ^ { \circ }\), the range is found to be 125 m .
3. Comment briefly on the accuracy of the standard projectile model in this situation. The company refines the model by assuming that the ball has a constant deceleration of \(2 \mathrm {~ms} ^ { - 2 }\) in the horizontal direction. In this new model, the resistance to the vertical motion is still neglected and so the flight time is still 4.08 s when the angle of projection is \(30 ^ { \circ }\).
4. Using the new model, with \(\alpha = 30 ^ { \circ }\), show that the horizontal displacement from the point of projection, \(x \mathrm {~m}\) at time \(t \mathrm {~s}\), is given by $$x = 40 t \cos 30 ^ { \circ } - t ^ { 2 }$$ Find the range and hence show that this new model is reasonably accurate in this case. The company then sets the angle of projection to \(45 ^ { \circ }\) while retaining a projection speed of \(40 \mathrm {~ms} ^ { - 1 }\). With this setting the range of the ball is found to be 135 m .
5. Investigate whether the new model is also accurate for this angle of projection.
6. Make one suggestion as to how the model could be further refined. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7bcde451-5c86-4ed6-b6f5-62c1ad77618c-2_722_1311_192_453} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows a platform 10 m long and 2 m high standing on horizontal ground. A small ball projected from the surface of the platform at one end, O , just misses the other end, P . The ball is projected at \(68.5 ^ { \circ }\) to the horizontal with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. At time \(t\) seconds after projection, the horizontal and vertical displacements of the ball from O are \(x \mathrm {~m}\) and \(y \mathrm {~m}\).