A ball, of mass \(m \mathrm {~kg}\), is moving with velocity \(( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( - 2 \mathbf { i } - 4 \mathbf { j } )\) Ns. Immediately after the impulse is applied, the ball has velocity \(( 3 \mathbf { i } + k \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the values of the constants \(k\) and \(m\).
A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \(( 12 t - 15 ) \mathrm { ms } ^ { - 2 }\). Find
the velocity of \(P\) at time \(t\) seconds after it leaves \(O\),
the value of \(t\) when the speed of \(P\) is \(36 \mathrm {~ms} ^ { - 1 }\).
A non-uniform ladder \(A B\), of length \(3 a\), has its centre of mass at \(G\), where \(A G = 2 a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(A B\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac { 14 } { 9 }\). Calculate the coefficient of friction between the ladder and the ground.