Impulse from velocity change

1. A particle \(P\) of mass 0.3 kg is moving with velocity \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The particle receives an impulse I Ns.
Immediately after receiving the impulse, the velocity of \(P\) is \(( 7 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the magnitude of \(\mathbf { I }\)
2. Find the angle between the direction of \(\mathbf { I }\) and the direction of motion of \(P\) immediately before receiving the impulse.

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular 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 \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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.

6. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(\mathbf { F } = \left( 1.5 t ^ { 2 } - 3 \right) \mathbf { i } + 2 t \mathbf { j }\). When \(t = 2\), the velocity of \(P\) is \(( - 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of \(P\) at time \(t\) seconds.
2. Show that, when \(t = 3\), the velocity of \(P\) is \(( 9 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf { Q }\) Ns. Immediately after the impulse the velocity of \(P\) is \(( - 3 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
3. the magnitude of \(\mathbf { Q }\),
4. the angle between \(\mathbf { Q }\) and \(\mathbf { i }\).

3. A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \(( - 30 \mathbf { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck, the velocity of the ball is \(( 16 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the magnitude of the impulse exerted on the ball by the bat. In the subsequent motion, the position vector of the ball is \(\mathbf { r }\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf { r } = \left[ 16 t \mathbf { i } + \left( 20 t - 5 t ^ { 2 } \right) \mathbf { j } \right]\). Using this model,
2. find the speed of the ball when \(t = 3\).

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse I Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the angle between \(\mathbf { I }\) and \(\mathbf { j }\).

1. An ice hockey puck of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity \(( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)

2
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude \(I \mathrm {~N} \mathrm {~s}\) at an angle of \(\theta\) to the horizontal. Immediately before the ball bounces it has speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal. Immediately after the ball bounces it has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal (see diagram). Find \(I\) and \(\theta\).

2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark]
\(M L T ^ { - 2 }\)
\(M L T ^ { - 1 }\)
MLT
\(M L T { } ^ { 2 }\)

3 A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \mathrm {~ms} ^ { - 1 }\)
The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \mathrm {~ms} ^ { - 1 }\)
Show that the impulse on the ball has magnitude 4 Ns
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