Vector impulse: find velocity or speed after impulse

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. Find the speed of \(P\) immediately after receiving the impulse.

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 2 \mathbf { i } - \mathbf { j } )\) Ns.

Show that the kinetic energy gained by \(P\) as a result of the impulse is 12 J .

1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.

Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.

A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by $$\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.$$

1. Calculate the speed of \(P\) when \(t = 3\). [5]

When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.

1. Find the velocity of \(P\) immediately after the impulse. [3]

A tennis ball of mass \(0.1\) kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30\mathbf{i}\) m s\(^{-1}\). The racquet exerts an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) N s on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit. [4]

The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass 0.1 kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).

1. Find the velocity of the ball immediately before it is hit. [3]

In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.

1. Find the greatest height of the ball above the ground in the subsequent motion. [3]

The ball is caught when it is again 1 m above the ground.

1. Find the distance from the point where the ball is hit to the point where it is caught. [4]

A ball of mass 0.5 kg is moving with velocity \(12\mathbf{i}\) m s\(^{-1}\) when it is struck by a bat. The impulse received by the ball is \((-4\mathbf{i} + 7\mathbf{j})\) N s. By modelling the ball as a particle, find

1. the speed of the ball immediately after the impact, [4]
2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf{i}\), [2]
3. the kinetic energy gained by the ball as a result of the impact. [2]

A ball, of mass \(m\) kg, is moving with velocity \((5\mathbf{i} - 3\mathbf{j})\) 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 \((3\mathbf{i} + k\mathbf{j})\) ms\(^{-1}\). Find the values of the constants \(k\) and \(m\). [6 marks]

A particle of mass 5 kg is moving with velocity \(2\mathbf{i} + 5\mathbf{j}\) m s\(^{-1}\). It receives an impulse of magnitude 15 N s in the direction \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\). Find the velocity of the particle immediately afterwards. [3]