Velocity after impulse (find unknown constant)

1. A particle \(P\) of mass 1.5 kg is moving with velocity \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of magnitude 15Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(\boldsymbol { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the two possible values of \(v\).

5. A particle of mass 0.5 kg is moving on a smooth horizontal surface with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(K ( \mathbf { i } + \mathbf { j } ) \mathrm { N } \mathrm { s }\), where \(K\) is a positive constant. Immediately after receiving the impulse the particle is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction which makes an acute angle \(\theta\) with the vector \(\mathbf { i }\). Find

1. the value of \(K\),
2. the size of angle \(\theta\).

3. A particle \(P\) of mass 0.5 kg is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(P\) receives an impulse of magnitude \(\sqrt { \frac { 5 } { 2 } } \mathrm { Ns }\) Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) Given that \(\lambda\) is a constant, find the two possible values of \(\lambda\)

2. A particle of mass 2 kg is moving with velocity \(3 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(( \lambda \mathbf { i } - 2 \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after the impulse is received, the speed of the particle is \(6 \mathrm {~ms} ^ { - 1 }\). Find the possible values of \(\lambda\).

4. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after \(P\) receives the impulse, the speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Given that \(\mathbf { J } = c ( - \mathbf { i } + 2 \mathbf { j } )\), where \(c\) is a constant, find the two possible values of \(c\).
(6)

1. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

The particle receives an impulse \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { Ns }\), where \(\lambda\) is a constant.
Immediately after receiving the impulse, the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the possible values of \(\lambda\)

1 A particle \(P\) of mass 0.3 kg is moving on a smooth horizontal surface with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. The magnitude of the impulse is 0.6 Ns .

1. (a) Find the greatest possible speed of \(P\) after the impulse acts.
   (b) Find the least possible speed of \(P\) after the impulse acts.
2. In fact the speed of \(P\) after the impulse acts is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle the impulse makes with the original direction of travel of \(P\) and draw a sketch to make this direction clear.