Projectile energy - basic KE/PE calculation

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]

1 A stone, of mass 0.4 kg , is thrown vertically upwards with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 6 metres above ground level.

1. Calculate the initial kinetic energy of the stone.
2. 1. Show that the kinetic energy of the stone when it hits the ground is 36.3 J , correct to three significant figures.
   2. Hence find the speed at which the stone hits the ground.
   3. State one assumption that you have made.

1 A ball is thrown vertically upwards from ground level with an initial speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has a mass of 0.6 kg . Assume that the only force acting on the ball after it is thrown is its weight.

1. Calculate the initial kinetic energy of the ball.
2. By using conservation of energy, find the maximum height above ground level reached by the ball.
3. By using conservation of energy, find the kinetic energy and the speed of the ball when it is at a height of 3 m above ground level.
4. State one modelling assumption which has been made.

1 A plane is dropping packets of aid as it flies over a flooded village. The speed of a packet when it leaves the plane is \(60 \mathrm {~ms} ^ { - 1 }\). The packet has mass 25 kg . The packet falls a vertical distance of 34 metres to reach the ground.

1. Calculate the kinetic energy of the packet when it leaves the plane.
2. Calculate the potential energy lost by the packet as it falls to the ground.
3. Assume that the effect of air resistance on the packet as it falls can be neglected.
   1. Find the kinetic energy of the packet when it reaches the ground.
   2. Hence find the speed of the packet when it reaches the ground.

1 Tim is playing cricket. He hits a ball at a point \(A\). The speed of the ball immediately after being hit is \(11 \mathrm {~ms} ^ { - 1 }\). The ball strikes a tree at a point \(B\). The height of \(B\) is 5 metres above the height of \(A\).
The ball is to be modelled as a particle of mass 0.16 kg being acted upon only by gravity.

1. Calculate the initial kinetic energy of the ball.
2. Calculate the potential energy gained by the ball as it moves from the point \(A\) to the point \(B\).
3. 1. Find the kinetic energy of the ball immediately before it strikes the tree.
   2. Hence find the speed of the ball immediately before it strikes the tree.

2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The rock falls a vertical distance of 51 metres to reach the surface of the sea.

1. Calculate the kinetic energy of the rock when it is thrown.
2. Calculate the potential energy lost by the rock when it reaches the surface of the sea.
3. 1. Find the kinetic energy of the rock when it reaches the surface of the sea.
   2. Hence find the speed of the rock when it reaches the surface of the sea.
4. State one modelling assumption which has been made.
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}

1 In an Olympic diving competition, Kim, who has mass 58 kg , dives from a fixed platform, 10 metres above the surface of the pool. She leaves the platform with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assume that Kim's weight is the only force that acts on her after she leaves the platform. Kim is to be modelled as a particle which is initially 1 metre above the platform.

1. Calculate Kim's initial kinetic energy.
2. By using conservation of energy, find Kim's speed when she is 6 metres below the platform.

1 Alan, of mass 76 kg , performed a ski jump. He took off at the point \(A\) at the end of the ski run with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and landed at the point \(B\). The level of the point \(B\) is 31 metres vertically below the level of the point \(A\), as shown in the diagram. Assume that his weight is the only force that acted on Alan during the jump. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-2_581_914_664_571}

1. Calculate the kinetic energy of Alan when he was at the point \(A\).
2. Calculate the potential energy lost by Alan during the jump as he moved from the point \(A\) to the point \(B\).
3. 1. Find the kinetic energy of Alan when he reached the point \(B\).
   2. Hence find the speed of Alan when he reached the point \(B\).

2 Carol, a circus performer, is on a swing. She jumps off the swing and lands in a safety net. When Carol leaves the swing, she has a speed of \(7 \mathrm {~ms} ^ { - 1 }\) and she is at a height of 8 metres above the safety net. Carol is to be modelled as a particle of mass 52 kg being acted upon only by gravity.

1. Find the kinetic energy of Carol when she leaves the swing.
2. Show that the kinetic energy of Carol when she hits the net is 5350 J , correct to three significant figures.
3. Find the speed of Carol when she hits the net.

1. A stone of mass 0.5 kg is projected vertically upwards with a speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(A\). The point \(A\) is 2.5 m above horizontal ground.

The speed of the stone as it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The motion of the stone from the instant it is projected from \(A\) until the instant it hits the ground is modelled as that of a particle moving freely under gravity.

1. Use the model and the principle of conservation of mechanical energy to find the value of \(U\). In reality, the stone will be subject to air resistance as it moves from \(A\) to the ground.
2. State how this would affect your answer to part (a). The ground is soft and the stone sinks a vertical distance \(d \mathrm {~cm}\) into the ground. The resistive force exerted on the stone by the ground is modelled as a constant force of magnitude 2000 N and the stone is modelled as a particle.
3. Use the model and the work-energy principle to find the value of \(d\), giving your answer to 3 significant figures.

2 A stone, of mass 6 kg , is thrown vertically upwards with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 4 metres above ground level.

1. Calculate the initial kinetic energy of the stone.
2. 1. Show that the kinetic energy of the stone when it hits the ground is 667 J , correct to three significant figures.
   2. Hence find the speed of the stone when it hits the ground.
   3. State two modelling assumptions that you have made.

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.

3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J
20 J

A ball of mass 0.2 kg is projected vertically upwards from a point \(O\) with speed 20 m s\(^{-1}\). The non-gravitational resistance acting on the ball is modelled as a force of constant magnitude 1.24 N and the ball is modelled as a particle. Find, using the work-energy principle, the speed of the ball when it first reaches the point which is 8 m vertically above \(O\). [6]

An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \text{ m s}^{-1}\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.

1. Calculate the kinetic energy of the salmon when it is dropped by the eagle. [2 marks]
2. Calculate the potential energy lost by the salmon as it falls to the sea. [2 marks]
3. 1. Find the kinetic energy of the salmon when it reaches the sea. [2 marks]
   2. Hence find the speed of the salmon when it reaches the sea. [2 marks]

A stone, of mass \(0.3\) kg, is thrown with a speed of \(8 \text{ m s}^{-1}\) from a point at a height of \(5\) metres above a horizontal surface.

1. Calculate the initial kinetic energy of the stone. [2 marks]
2. 1. Find the kinetic energy of the stone when it hits the surface. [3 marks]
   2. Hence find the speed of the stone when it hits the surface. [2 marks]
   3. State one modelling assumption that you have made. [1 mark]

In a school experiment, a particle, of mass \(m\) kilograms, is released from rest at a point \(h\) metres above the ground. At the instant it reaches the ground, the particle has velocity \(v \text{ m s}^{-1}\)

1. Show that $$v = \sqrt{2gh}$$ [2 marks]
2. A student correctly used \(h = 18\) and measured \(v\) as 20 The student's teacher claims that the machine measuring the velocity must have been faulty. Determine if the teacher's claim is correct. Fully justify your answer. [3 marks]

The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below. The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water. \includegraphics{figure_2} When the woman dives, she projects herself from the platform with a speed of \(7.8\text{ ms}^{-1}\).

1. Find the kinetic energy of the woman when she leaves the platform. [2]
2. Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of \(16\text{ ms}^{-1}\). [6]
3. Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be \(13\text{ ms}^{-1}\). Determine the amount of energy lost to air resistance according to the refined model. [3]