Show that the speed of \(P\) when it first begins to move in a circle is \(\sqrt { 3 g }\).
In the subsequent motion, when the string first makes an angle of \(45 ^ { \circ }\) with the downwards vertical,
calculate the speed \(v\) of \(P\),
determine the tension in the string.
\item At time \(t = 0 \mathrm {~s}\), the position vector of an object \(A\) is \(\mathbf { i } \mathrm { m }\) and the position vector of another object \(B\) is \(3 \mathbf { i } \mathrm {~m}\). The constant velocity vector of \(A\) is \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } \mathrm {~ms} ^ { - 1 }\) and the constant velocity vector of \(B\) is \(\mathbf { i } + 3 \mathbf { j } - 5 \mathbf { k } \mathrm {~ms} ^ { - 1 }\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\).
\item Relative to a fixed origin \(O\), the position vector \(\mathbf { r } \mathrm { m }\) at time \(t \mathrm {~s}\) of a particle \(P\), of mass 0.4 kg , is given by
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$$\mathbf { r } = \mathrm { e } ^ { 2 t } \mathbf { i } + \sin ( 2 t ) \mathbf { j } + \cos ( 2 t ) \mathbf { k }$$
Show that the velocity vector \(\mathbf { v }\) and the position vector \(\mathbf { r }\) are never perpendicular to each other.
Given that the speed of \(P\) at time \(t\) is \(v _ { \mathrm { ms } } ^ { - 1 }\), show that
$$v ^ { 2 } = 4 \mathrm { e } ^ { 4 t } + 4$$
Find the kinetic energy of \(P\) at time \(t\).
Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\).
Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\).