7 A particle \(P\) is projected from a fixed point \(O\) on a straight line. The displacement \(x\) m of \(P\) from \(O\) at time \(t \mathrm {~s}\) after projection is given by \(x = 0.1 t ^ { 3 } - 0.3 t ^ { 2 } + 0.2 t\).
- Express the velocity and acceleration of \(P\) in terms of \(t\).
- Show that when the acceleration of \(P\) is zero, \(P\) is at \(O\).
- Find the values of \(t\) when \(P\) is stationary.
At the instant when \(P\) first leaves \(O\), a particle \(Q\) is projected from \(O\). \(Q\) moves on the same straight line as \(P\) and at time \(t \mathrm {~s}\) after projection the velocity of \(Q\) is given by \(\left( 0.2 t ^ { 2 } - 0.4 \right) \mathrm { ms } ^ { - 1 } . P\) and \(Q\) collide first when \(t = T\).
- Show that \(T\) satisfies the equation \(t ^ { 2 } - 9 t + 18 = 0\), and hence find \(T\).