5. \begin{figure}[h] \end{figure} At time \(t = 0\), a small stone is projected with velocity \(35 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The stone is projected at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)
In an initial model

• the stone is modelled as a particle \(P\) moving freely under gravity
• the stone hits the ground at the point \(A\)

Figure 4 shows the path of \(P\) from \(O\) to \(A\).
For the motion of \(P\) from \(O\) to \(A\)

• at time \(t\) seconds, the horizontal distance of \(P\) from \(O\) is \(x\) metres
• at time \(t\) seconds, the vertical distance of \(P\) above the ground is \(y\) metres
  1. Using the model, show that

$$y = \frac { 3 } { 4 } x - \frac { 1 } { 160 } x ^ { 2 }$$

Use the answer to (a), or otherwise, to find the length \(O A\). Using the model, the greatest height of the stone above the ground is found to be \(H\) metres.

Use the answer to (a), or otherwise, to find the value of \(H\).

• The model is refined to include air resistance.

Using this new model, the greatest height of the stone above the ground is found to be \(K\) metres.

State which is greater, \(H\) or \(K\), justifying your answer.

State one limitation of this refined model.