Energy change from impulse

1. A particle \(P\) of mass 0.6 kg is moving with velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { I } \mathrm { N }\) s. Immediately after receiving the impulse, \(P\) has velocity ( \(2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the kinetic energy lost by \(P\) as a result of receiving the 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 0.3 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Find the change in the kinetic energy of the particle due to the impulse.
   (6)
2. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 27 - 3 t ^ { 2 } \right) \mathbf { i } + \left( 8 - t ^ { 3 } \right) \mathbf { j }$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\) relative to a fixed origin \(O\), where \(\mathbf { r } = - 5 \mathbf { i } + 2 \mathbf { j }\) Find

1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf { i }\),
2. the position vector of \(P\) at the instant when \(t = 3\)

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.
(5)
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1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the magnitude of the impulse of the bat on the ball,
2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
3. the kinetic energy lost by the ball in the impact.

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

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

1. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { Ns }\). The angle between the velocity of \(P\) before the impulse and the velocity of \(P\) after the impulse is \(\theta ^ { \circ }\).

Find

1. the value of \(\theta\),
2. the kinetic energy gained by \(P\) as a result of the impulse.