Impulse from variable force

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

6 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 \(28 \mathrm {~ms} ^ { - 1 }\) Immediately after the impact, the ball is travelling horizontally with speed \(14 \mathrm {~ms} ^ { - 1 }\) in the opposite direction. 6

1. Find the magnitude of the impulse exerted by the bat on the ball.
   [0pt] [2 marks]
   6
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 $$F = 10 k t ( 0.05 - t )$$ where \(k\) is a constant.
   Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\)
   [0pt] [3 marks]
   \(7 \quad\) Use \(g\) as \(9.81 \mathrm {~ms} ^ { - 2 }\) in this question. A light elastic string has one end attached to a fixed point \(A\) on a smooth plane inclined at \(25 ^ { \circ }\) to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg , which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons.
   The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from \(A\) and then released.

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, \(F\) newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where \(k\) and \(b\) are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

1. State the value of \(b\).
   7
2. Find the magnitude of the impulse on the disc, giving your answer in terms of \(k\).
   7
3. The disc is travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the wall.
   The disc rebounds with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   Find \(k\).
   [0pt] [3 marks]

3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F \mathrm {~N}\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).

3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.

1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
2. Find, to two significant figures, the speed of the ball immediately after the impact.
3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.

3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).

1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
2. Find the speed of \(P\) when \(t = 3\).
3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).

1 A ball of mass 0.2 kg is hit directly by a bat. Just before the impact, the ball is travelling horizontally with speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Just after the impact, the ball is travelling horizontally with speed \(32 \mathrm {~ms} ^ { - 1 }\) in the opposite direction.

1. Find the magnitude of the impulse exerted on the ball.
2. At time \(t\) seconds after the ball first comes into contact with the bat, the force exerted by the bat on the ball is \(k \left( 0.9 t - 10 t ^ { 2 } \right)\) newtons, where \(k\) is a constant and \(0 \leqslant t \leqslant 0.09\). The bat stays in contact with the ball for 0.09 seconds. Find the value of \(k\).

1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time \(t\) seconds after coming into contact with the wall, the force exerted by the wall on the player is \(4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )\) newtons, where \(0 \leqslant t \leqslant 0.5\).

1. Find the magnitude of the impulse exerted by the wall on the player.
2. The player rebounds from the wall. Find the player's speed immediately after the collision.

3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time \(t\) seconds has magnitude \(( 3 t + 1 ) \mathrm { N }\). When \(t = 0\), the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse of the force on the particle between the times \(t = 0\) and \(t = 3\).
2. Hence find the velocity of the particle when \(t = 3\).
3. Find the value of \(t\) when the velocity of the particle is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 At a fairground a dodgem car is moving in a straight horizontal line towards a side wall that is perpendicular to the velocity of the car. The speed of the car is \(1.8 \mathrm {~ms} ^ { - 1 }\)
It collides with the side wall and rebounds along its original path with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total mass of the dodgem car and the passengers is 250 kg
6

1. Find the magnitude of the impulse on the car during the collision with the side wall.
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2. A possible model for the magnitude of the force, \(F\) newtons, acting on the dodgem car due to its collision with the side wall is given by $$F = k t ( 4 - 5 t ) \quad \text { for } 0 \leq t \leq 0.8$$ 6
3. 1. Find the value of \(k\).
4. (ii) Determine the maximum magnitude of the force predicted by the model. 6
5. (ii) Determine the maximum magnitude of the fored bed bed at

6 An ice hockey puck, of mass 0.2 kg , is moving in a straight line on a horizontal ice rink under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude ( \(2 t + 3\) ) newtons.
The force acts on the puck from \(t = 0\) to \(t = T\)
6

1. Show that the magnitude of the impulse of the force is \(a T ^ { 2 } + b T\), where \(a\) and \(b\) are integers to be found.
   [0pt] [3 marks]
   6
2. While the force acts on the puck, its speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Use your answer from part (a) to find \(T\), giving your answer to three significant figures.
   Fully justify your answer.

7 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, \(F = 6 \mathrm { e } ^ { t } + 2 \mathrm { e } ^ { 2 t }\) where \(0 \leq t \leq \ln 8\) 7

1. Find the impulse of \(F\) over the interval \(0 \leq t \leq \ln 8\)
   7
2. The particle has a mass of 2 kg and at time \(t = 0\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the velocity of the particle when \(t = \ln 8\)