6.02d Mechanical energy: KE and PE concepts

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.

3 A pump is being used to empty a flooded basement.
In one minute, 400 litres of water are pumped out of the basement.
The water is raised 8 metres and is ejected through a pipe at a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
The mass of 400 litres of water is 400 kg .

1. Calculate the gain in potential energy of the 400 litres of water.
2. Calculate the gain in kinetic energy of the 400 litres of water.
3. Hence calculate the power of the pump, giving your answer in watts.

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 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.

1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the initial kinetic energy of the box.
2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
3. Hence find the speed of the box when it hits the ground.
4. State two modelling assumptions which you have made.

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.

3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.

1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
   1. Find the kinetic energy of Simon when he reaches the point \(R\).
   2. Hence find the speed of Simon when he reaches the point \(R\).
2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
   Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

3. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. At a certain instant, a particle \(P\) of mass 1.8 kg is moving with velocity \(( 24 \mathrm { i } - 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\).

1. Calculate the kinetic energy of \(P\) at this instant. \(P\) is now subjected to a constant retardation. After 10 seconds, the velocity of \(P\) is \(( - 12 \mathbf { i } + 3 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
2. Calculate the work done by the retarding force over the 10 seconds.

4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.

1. Find, in J , the change in the potential energy of the block as it moves down the plane.
2. Hence find the distance travelled by the block down the plane.
3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}

5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find

1. the increase in kinetic energy,
2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
3. Calculate the distance \(A B\).

6 \begin{figure}[h] \end{figure} A container is constructed from a hollow cylindrical shell and a hollow cone which are joined along their circumferences. The cylindrical shell has radius 0.2 m , and the cone has semi-vertical angle \(30 ^ { \circ }\). Two identical small spheres \(P\) and \(Q\) move independently in horizontal circles on the smooth inner surface of the container (see Fig. 1). Each sphere has mass 0.3 kg .

1. \(P\) moves in a circle of radius 0.12 m and is in contact with only the conical part of the container. Calculate the angular speed of \(P\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_278_209_1845_1009} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(Q\) moves with speed \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is in contact with both the cylindrical and conical surfaces of the container (see Fig. 2). Calculate the magnitude of the force which the cylindrical shell exerts on the sphere.
3. Calculate the difference between the mechanical energy of \(P\) and of \(Q\). \section*{[Question 7 is printed overleaf.]}

5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 4 m above the level of \(A\).

1. Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
2. Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance \(A B\). At \(B\) the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
3. Find the acceleration with which the cyclist starts to move horizontally.

2 A car of mass 1200 kg travels along a road for two minutes during which time it rises a vertical distance of 60 m and does \(1.8 \times 10 ^ { 6 } \mathrm {~J}\) of work against the resistance to its motion. The speeds of the car at the start and at the end of the two minutes are the same.

1. Calculate the average power developed over the two minutes. The car now travels along a straight level road at a steady speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while developing constant power of 13.5 kW .
2. Calculate the resistance to the motion of the car. How much work is done against the resistance when the car travels 200 m ? While travelling at \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car starts to go down a slope inclined at \(5 ^ { \circ }\) to the horizontal with the power removed and its brakes applied. The total resistance to its motion is now 1500 N .
3. Use an energy method to determine how far down the slope the car travels before its speed is halved. Suppose the car is travelling along a straight level road and developing power \(P \mathrm {~W}\) while travelling at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of \(R \mathrm {~N}\).
4. Show that \(P = ( R + 1200 a ) v\) and deduce that if \(P\) and \(R\) are constant then if \(a\) is not zero it cannot be constant.

1 A bus of mass 8 tonnes is driven up a hill on a straight road. On one part of the hill, the power of the driving force on the bus is constant at 20 kW for one minute.

1. Calculate how much work is done by the driving force in this time. During this minute the speed of the bus increases from \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\) and, in addition to the work done against gravity, 125000 J of work is done against the resistance to motion of the bus parallel to the slope.
2. Calculate the change in the kinetic energy of the bus.
3. Calculate the vertical displacement of the bus. On another stretch of the road, a driving force of power 26 kW is required to propel the bus up a slope of angle \(\theta\) to the horizontal at a constant speed of \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), against a resistance to motion of 225 N parallel to the slope.
4. Calculate the angle \(\theta\). The bus later travels up the same slope of angle \(\theta\) to the horizontal at the same constant speed of \(6.5 \mathrm {~ms} ^ { - 1 }\) but now against a resistance to motion of 155 N parallel to the slope.
5. Calculate the power of the driving force on the bus.

4

1. A large nail of mass 0.02 kg has been driven a short distance horizontally into a fixed block of wood, as shown in Fig. 4.1, and is to be driven horizontally further into the block. The wood produces a constant resistance of 2.43 N to the motion of the nail. The situation is modelled by assuming that linear momentum is conserved when the nail is struck, that all the impacts with the nail are direct and that the head of the nail never reaches the wood. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_279_711_482_676} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The nail is first struck by an object of mass 0.1 kg that is moving parallel to the nail with linear momentum of magnitude 0.108 Ns . The object becomes firmly attached to the nail.
   1. Calculate the speed of the nail and object immediately after the impact.
   2. Calculate the time for which the nail and object move, and the distance they travel in that time. On a second attempt to drive in the nail, it is struck by the same object of mass 0.1 kg moving parallel to the nail with the same linear momentum of magnitude 0.108 Ns . This time the object does not become attached to the nail and after the contact is still moving parallel to the nail. The coefficient of restitution in the impact is \(\frac { 1 } { 3 }\).
   3. Calculate the speed of the nail immediately after this impact.
2. A small ball slides on a smooth horizontal plane and bounces off a smooth straight vertical wall. The speed of the ball is \(u\) before the impact and, as shown in Fig. 4.2, the impact turns the path of the ball through \(90 ^ { \circ }\). The coefficient of restitution in the collision between the ball and the wall is \(e\). Before the collision, the path is inclined at \(\alpha\) to the wall. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_294_590_1804_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Write down, in terms of \(u , e\) and \(\alpha\), the components of the velocity of the ball parallel and perpendicular to the wall before and after the impact.
   2. Show that \(\tan \alpha = \frac { 1 } { \sqrt { e } }\).
   3. Hence show that \(\alpha \geqslant 45 ^ { \circ }\).

1

1. A stone of mass 0.6 kg falls vertically 1.5 m from A to B against resistance. Its downward speeds at A and \(B\) are \(5.5 \mathrm {~ms} ^ { - 1 }\) and \(7.5 \mathrm {~ms} ^ { - 1 }\) respectively.
   1. Calculate the change in kinetic energy and the change in gravitational potential energy of the stone as it falls from A to B .
   2. Calculate the work done against resistance to the motion of the stone as it falls from A to B .
   3. Assuming the resistive force is constant, calculate the power with which the resistive force is retarding the stone when it is at A .
2. A uniform plank is inclined at \(40 ^ { \circ }\) to the horizontal. A box of mass 0.8 kg is on the point of sliding down it. The coefficient of friction between the box and the plank is \(\mu\).
   1. Show that \(\mu = \tan 40 ^ { \circ }\). The plank is now inclined at \(20 ^ { \circ }\) to the horizontal.
   2. Calculate the work done when the box is pushed 3 m up the plank, starting and finishing at rest.

4

1. A small heavy object of mass 10 kg travels the path ABCD which is shown in Fig. 4. ABCD is in a vertical plane; CD and AEF are horizontal. The sections of the path AB and CD are smooth but section BC is rough. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-5_368_1323_402_338} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} You should assume that
   • the object does not leave the path when travelling along ABCD and does not lose energy when changing direction
   • there is no air resistance.
   Initially, the object is projected from A at a speed of \(16.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope.
   1. Show that the object gets beyond B . The section of the path BC produces a constant resistance of 14 N to the motion of the object.
   2. Using an energy method, find the velocity of the object at D . At D , the object leaves the path and bounces on the smooth horizontal ground between E and F , shown in Fig. 4. The coefficient of restitution in the collision of the object with the ground is \(\frac { 1 } { 2 }\).
   3. Calculate the greatest height above the ground reached by the object after its first bounce.
2. A car of mass 1500 kg travelling along a straight, horizontal road has a steady speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its driving force has power \(P \mathrm {~W}\). When at this speed, the power is suddenly reduced by \(20 \%\). The resistance to the car's motion, \(F \mathrm {~N}\), does not change and the car begins to decelerate at \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the values of \(P\) and \(F\).

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .

2. A particle \(P\) of mass 3 kg moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin, \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - 3 t \right) \mathbf { i } + \frac { 1 } { 6 } t ^ { 3 } \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find the velocity of \(P\) when \(t = 0\).
2. Find the kinetic energy lost by \(P\) in the interval \(0 \leq t \leq 2\).

6. The engine of a car of mass 1200 kg is working at a constant rate of 90 kW as the car moves along a straight horizontal road. The resistive forces opposing the motion of the car are constant and of magnitude 1800 N .

1. Find the acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find, in kJ, the kinetic energy of the car when it is travelling at maximum speed. The car ascends a hill which is straight and makes an angle \(\alpha\) with the horizontal. The power output of the engine and the non-gravitational forces opposing the motion remain the same. Given that the car can attain a maximum speed of \(25 \mathrm {~ms} ^ { - 1 }\),
3. find, in degrees correct to one decimal place, the value of \(\alpha\).
   (5 marks)