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

In a safety test, a car of mass 800 kg is driven directly at a wall at a constant speed of 15 m s\(^{-1}\). The constant force exerted by the wall on the car in bringing it to rest is 60 kN.

1. Calculate the magnitude of the impulse exerted by the wall on the car. [2 marks]
2. Find the time it takes for the car to come to rest. [2 marks]
3. Show that the deceleration of the car is 75 m s\(^{-2}\). [3 marks]

A ball of mass 0.15 kg is hit directly by a vertical cricket bat. Immediately before the impact, the ball is travelling horizontally with speed 28 m s\(^{-1}\) Immediately after the impact, the ball is travelling horizontally with speed 14 m s\(^{-1}\) in the opposite direction.

1. Find the magnitude of the impulse exerted by the bat on the ball. [2 marks]
2. In a simple model the force, \(F\) newtons, exerted by the bat on the ball, \(t\) seconds after the initial impact, is given by \(F = 10kt (0.05 - t)\) where \(k\) is a constant. Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\) [3 marks]

A single force, \(F\) newtons, acts on a particle moving on a straight, smooth, horizontal line. The force \(F\) acts in the direction of motion of the particle. At time \(t\) seconds, \(F = 6e^t + 2e^{2t}\) where \(0 \leq t \leq \ln 8\)

1. Find the impulse of \(F\) over the interval \(0 \leq t \leq \ln 8\) [2 marks]
2. The particle has a mass of 2 kg and at time \(t = 0\) has velocity \(5 \text{ m s}^{-1}\) Find the velocity of the particle when \(t = \ln 8\) [3 marks]

As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac{t}{5}\) Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\) Circle your answer. [1 mark] 0.3 N s \quad 0.6 N s \quad 0.9 N s \quad 1.8 N s