Velocity after impulse (direct calculation)

1. A particle of mass 4 kg is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( 7 \mathbf { i } - 5 \mathbf { j } )\) Ns.

Find the speed of the particle immediately after receiving the impulse.

1. \begin{figure}[h] \end{figure} A particle, \(P\), of mass 0.8 kg , moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane, receives a horizontal impulse of magnitude 6 N s. The angle between the initial direction of motion of \(P\) and the direction of the impulse is \(50 ^ { \circ }\), as shown in Figure 1. Find the speed of \(P\) immediately after receiving the impulse.

2. A particle \(P\) of mass 0.75 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = \left( t ^ { 2 } + 2 \right) \mathbf { i } - 6 t \mathbf { j }$$

1. Find the magnitude of \(\mathbf { F }\) when \(t = 4\).
   (5) When \(t = 5\), the particle \(P\) receives an impulse of magnitude \(9 \sqrt { } 2 \mathrm { Ns }\) in the direction of the vector \(\mathbf { i } - \mathbf { j }\).
2. Find the velocity of \(P\) immediately after the impulse.

1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,

$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).

1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
2. Find the speed of \(P\) immediately after it receives the impulse.

1. A particle of mass 0.25 kg is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives the impulse \(( 5 \mathbf { i } - 3 \mathbf { j } )\) N s.

Find the speed of the particle immediately after the impulse.

1. A particle \(P\) of mass 2 kg is moving with velocity \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 3 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\).

Find the speed of \(P\) immediately after the impulse is applied.
(5)

1 A smooth horizontal surface lies in the \(x - y\) plane. A particle \(P\) of mass 0.5 kg is moving on the surface with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(x\)-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the \(x\)-direction and \(y\)-direction respectively.

1. Find the components in the \(x\)-direction and the \(y\)-direction of the velocity of \(P\) immediately after the blow. Hence show that the speed of \(P\) immediately after the blow is \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(P\) is struck by a second horizontal blow whose impulse is \(\mathbf { I }\).
2. Given that \(P\) 's direction of motion immediately after this blow is parallel to the \(x\)-axis, write down the component of \(\mathbf { I }\) in the \(y\)-direction.

A ball of mass \(2\)kg is moving with velocity \((3\mathbf{i} - 2\mathbf{j})\)ms\(^{-1}\) when it is struck by a bat. The impulse on the ball is \((-8\mathbf{i} + 10\mathbf{j})\)Ns.

1. Find the speed of the ball immediately after the impact. [4]
2. State one modelling assumption you have used in answering part (a). [1]