A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after \(3 \cdot 3\) seconds.
Calculate the height above ground from which the lead was dropped.
According to a simple model, the ball hits the ground at the same time as the lead.
State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution.
The piece of lead is now thrown again from \(P\), with speed \(7 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal, as shown.
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Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown.
Deduce that \(y = \frac { \sqrt { } 3 } { 3 } x - \frac { 2 } { 15 } x ^ { 2 }\).
Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\).