Calculate impulse from force-time data

1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A particle \(P\) has mass 10 kg and a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(4 \mathbf { i } + 3 \mathbf { j }\). A force of \(( - 4 \mathbf { i } + 15 \mathbf { j } ) \mathrm { N }\) acts on P for 8 seconds.

1. Calculate the impulse of the force over the 8 seconds.
2. Hence find the speed of P at the end of the 8 seconds.

4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude \(\left( t ^ { 3 } + t \right) \mathrm { N }\), where \(t\) is the time, in seconds, for which the force has been acting.

1. Find the magnitude of the impulse exerted by the force on the particle between the times \(t = 0\) and \(t = 4\).
2. Hence find the speed of the particle when \(t = 4\).
3. Find the time taken for the particle to reach a speed of \(12 \mathrm {~ms} ^ { - 1 }\).

1 The graph shows how a force, \(F\) newtons, varies during a 5 second period of time.
\includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-02_575_1182_680_429} Calculate the magnitude of the impulse of the force.
Circle your answer.
[0pt] [1 mark]
17.5 N s
25 Ns
35 Ns
70 Ns