Energy change from impulse

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) The particle receives an impulse \(( - 2 \mathbf { i } + \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after receiving the impulse, the velocity of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The kinetic energy gained by \(P\) as a result of receiving the impulse is 22 J .

Find the possible values of \(\lambda\).

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

Find

1. the speed of the particle immediately after receiving the impulse,
2. the kinetic energy gained by the particle as a result of the impulse.

2. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( - 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { N }\) s. Find the kinetic energy of the particle immediately after receiving the impulse.
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2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] A ball has mass 0.2 kg . It is moving with velocity ( 30 i ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. The bat exerts an impulse of \(( - 4 \mathbf { i } + 4 \mathbf { j } )\) Ns on the ball. Find

1. the velocity of the ball immediately after the impact,
2. the angle through which the ball is deflected as a result of the impact,
3. the kinetic energy lost by the ball in the impact.

6 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

A particle of mass 0.3 kg is moving with velocity \((5\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) when it receives an impulse \((-3\mathbf{i} + 3\mathbf{j})\) N s. Find the change in the kinetic energy of the particle due to the impulse. [6]