Work done against air resistance - vertical motion

1 A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the ground is \(12 \mathrm {~ms} ^ { - 1 }\). Find the work done against air resistance.

1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.

2 A ball of mass 0.6 kg is thrown vertically upwards from ground level with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the initial kinetic energy of the ball.
2. Assuming that no resistance forces act on the ball, use an energy method to find the maximum height reached by the ball.
3. An experiment is conducted to confirm the maximum height for the ball calculated in part (b). In this experiment the ball rises to a height of only 8 metres.
   1. Find the work done against the air resistance force that acts on the ball as it moves.
   2. Assuming that the air resistance force is constant, find its magnitude.
4. Explain why it is not realistic to model the air resistance as a constant force.

4. A small stone, of mass 600 grams, is released from rest a height of 2 metres above ground level and falls under gravity. The time it takes to reach the ground is \(T\) seconds. The stone is then again released from rest at the surface of a tank containing a 2 metre depth of liquid and reaches the bottom after \(2 T\) seconds. It may be assumed that the resisting force acting on the stone is constant.

1. Find the magnitude of the resisting force exerted on the stone by the liquid.
2. Find the speed with which the stone hits the bottom of the tank.

4 A child throws a ball of mass \(m \mathrm {~kg}\) vertically upwards with a speed of \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball leaves the child's hand at a height of 1.6 m above horizontal ground.

1. Ignoring any possible air resistance, use an energy method to determine the maximum height reached by the ball above the ground. In fact, the ball only reaches a height of 4.1 m above the ground. For the rest of this question you should assume that the air resistance may be modelled as a constant force acting in the opposite direction to the ball's motion.
2. Show that the ball does 0.568 mJ of work against air resistance per metre travelled.
3. Calculate the speed of the ball just before it hits the ground. The ball bounces off the ground and first comes instantaneously to rest 2.8 m above the ground.
4. Determine the coefficient of restitution between the ball and the ground. In the first impact between the ball and the ground, the magnitude of the impulse exerted on the ball by the ground is 12 Ns .
5. Determine the value of \(m\).

In this question use \(g = 9.8 \text{ m s}^{-2}\) A ball of mass 0.5 kg is projected vertically upwards with a speed of \(10 \text{ m s}^{-1}\)

1. Calculate the initial kinetic energy of the ball. [1 mark]
2. Assuming that the weight is the only force acting on the ball, use an energy method to show that the maximum height reached by the ball is approximately 5.1 m above the point of projection. [2 marks]
3. 1. A student conducts an experiment to verify the accuracy of the result obtained in part (b). They observe that the ball rises to a height of 4.4 m above the point of projection and concludes that this height difference is due to a resistance force, \(R\) newtons. Find the total work done against \(R\) whilst the ball is moving upwards. [2 marks]
   2. Using a model that assumes \(R\) is constant, find the magnitude of \(R\) [2 marks]
   3. Comment on the validity of the model used in part (c)(ii). [1 mark]