6.02d Mechanical energy: KE and PE concepts

5 A block \(B\) of mass 4 kg is pushed up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal by a force applied to \(B\), acting in the direction of motion of \(B\). The block passes through points \(P\) and \(Q\) with speeds \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(P\) and \(Q\) are 10 m apart with \(P\) below the level of \(Q\).

1. Find the decrease in kinetic energy of the block as it moves from \(P\) to \(Q\).
2. Hence find the work done by the force pushing the block up the slope as the block moves from \(P\) to \(Q\).
3. At the instant the block reaches \(Q\), the force pushing the block up the slope is removed. Find the time taken, after this instant, for the block to return to \(P\).

1 A particle of mass 0.6 kg is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. Use an energy method to find the speed of the particle after it has moved 15 m down the plane.

5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .

1. Find the power developed by the cyclist.
   The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
2. Find the change in kinetic energy.
3. Use an energy method to find \(d\). \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

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.

2 Two particles \(A\) and \(B\), of masses 3.2 kg and 2.4 kg respectively, lie on a smooth horizontal table. \(A\) moves towards \(B\) with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and collides with \(B\), which is moving towards \(A\) with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the two particles come to rest.

1. Find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-03_61_1569_495_328}
2. Find the loss of kinetic energy of the system due to the collision.

7 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-10_525_885_264_625} The diagram shows the vertical cross-section \(X Y Z\) of a rough slide. The section \(Y Z\) is a straight line of length 2 m inclined at an angle of \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The section \(Y Z\) is tangential to the curved section \(X Y\) at \(Y\), and \(X\) is 1.8 m above the level of \(Y\). A child of mass 25 kg slides down the slide, starting from rest at \(X\). The work done by the child against the resistance force in moving from \(X\) to \(Y\) is 50 J .

1. Find the speed of the child at \(Y\).
   It is given that the child comes to rest at \(Z\).
2. Use an energy method to find the coefficient of friction between the child and \(Y Z\), giving your answer as a fraction in its simplest form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A train of mass 180000 kg ascends a straight hill of length 1.5 km , inclined at an angle of \(1.5 ^ { \circ }\) to the horizontal. As it ascends the hill, the total work done to overcome the resistance to motion is 12000 kJ and the speed of the train decreases from \(45 \mathrm {~ms} ^ { - 1 }\) to \(40 \mathrm {~ms} ^ { - 1 }\). Find the work done by the engine of the train as it ascends the hill, giving your answer in kJ .

5 A straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.

1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.

7 A particle \(P\) of mass 0.2 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~ms} ^ { - 1 }\).

1. Show that the speed of \(P\) when it reaches 20 m above the ground is \(15 \mathrm {~ms} ^ { - 1 }\).
   When \(P\) reaches 20 m above the ground it collides with a second particle \(Q\) of mass 0.1 kg which is moving downwards at \(20 \mathrm {~ms} ^ { - 1 } . P\) is brought to instantaneous rest in the collision.
2. Find the velocity of \(Q\) immediately after the collision.
   When \(P\) reaches the ground it rebounds back directly upwards with half of the speed that it had immediately before hitting the ground.
3. Find the height above the ground at which \(P\) and \(Q\) next collide.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).

1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).

2 A box of mass 5 kg is pulled at a constant speed a distance of 15 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.

1. Find the work done against friction.
2. Find the change in gravitational potential energy of the box.
3. Find the work done by the pulling force.

4 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and \(m \mathrm {~kg}\) respectively, lie on a smooth horizontal plane. Initially, sphere \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collision \(A\) moves with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the two possible values of the loss of kinetic energy due to the collision.

2 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses km kg and \(m \mathrm {~kg}\) respectively, where \(k > 1\), are free to move on a smooth horizontal plane. \(A\) is moving towards \(B\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(A\) and \(B\) coalesce and move with speed \(4 \mathrm {~ms} ^ { - 1 }\).

1. Find \(k\).
2. Find, in terms of \(m\), the loss of kinetic energy due to the collision.

5 A car of mass 1600 kg travels at constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle of \(\sin ^ { - 1 } 0.12\) to the horizontal.

1. Find the change in potential energy of the car in 30 s .
2. Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.
3. Calculate, in kW , the power developed by the engine of the car.
4. Given that this power is suddenly decreased by \(15 \%\), find the instantaneous deceleration of the car.

3 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-04_416_792_260_674} The diagram shows a semi-circular track \(A B C\) of radius 1.8 m which is fixed in a vertical plane. The points \(A\) and \(C\) are at the same horizontal level and the point \(B\) is at the bottom of the track. The section \(A B\) is smooth and the section \(B C\) is rough. A small block is released from rest at \(A\).

1. Show that the speed of the block at \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of \(B\). The work done against the resistance force during the motion of the block from \(B\) to this point is 4.5 J .
2. Find the mass of the block.

3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.

1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.

2 A block of mass 20 kg is held at rest at the top of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The block is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane. There is a resistance force acting on the block. As the block moves 2 m down the plane from its point of projection, the work done against this resistance force is 50 J . Find the speed of the block when it has moved 2 m down the plane. \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-04_2716_38_109_2012}

7 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section \(A B C D\) of a surface. The parts \(A B\) and \(C D\) are straight and have lengths 2.5 m and 5.2 m respectively. \(A D\) is horizontal, and \(A B\) is inclined at \(60 ^ { \circ }\) to the horizontal. The points \(B\) and \(C\) are at the same height above \(A D\). The parts of the surface containing \(A B\) and \(B C\) are smooth. A particle \(P\) is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction \(A B\), and it subsequently reaches \(D\). The particle does not lose contact with the surface during this motion.

1. Find the speed of \(P\) at \(B\).
2. Show that the maximum height of the cross-section, above \(A D\), is less than 3.2 m .
3. State briefly why \(P\) 's speed at \(C\) is the same as its speed at \(B\).
4. The frictional force acting on the particle as it travels from \(C\) to \(D\) is 1.4 N . Given that the mass of \(P\) is 0.4 kg , find the speed with which \(P\) reaches \(D\).

4 The top of an inclined plane is at a height of 0.7 m above the bottom. A block of mass 0.2 kg is released from rest at the top of the plane and slides a distance of 2.5 m to the bottom. Find the kinetic energy of the block when it reaches the bottom of the plane in each of the following cases:

1. the plane is smooth,
2. the coefficient of friction between the plane and the block is 0.15 .

6 A block of mass 50 kg is pulled up a straight hill and passes through points \(A\) and \(B\) with speeds \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 200 m and \(B\) is 15 m higher than \(A\). For the motion of the block from \(A\) to \(B\), find

1. the loss in kinetic energy of the block,
2. the gain in potential energy of the block. The resistance to motion of the block has magnitude 7.5 N.
3. Find the work done by the pulling force acting on the block. The pulling force acting on the block has constant magnitude 45 N and acts at an angle \(\alpha ^ { \circ }\) upwards from the hill.
4. Find the value of \(\alpha\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_223_1456_1493_347} A lorry of mass 12500 kg travels along a road that has a straight horizontal section \(A B\) and a straight inclined section \(B C\). The length of \(B C\) is 500 m . The speeds of the lorry at \(A , B\) and \(C\) are \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. The work done against the resistance to motion of the lorry, as it travels from \(A\) to \(B\), is 5000 kJ . Find the work done by the driving force as the lorry travels from \(A\) to \(B\).
2. As the lorry travels from \(B\) to \(C\), the resistance to motion is 4800 N and the work done by the driving force is 3300 kJ . Find the height of \(C\) above the level of \(A B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552} \(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find

1. the kinetic energy of \(P\) at \(A\),
2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).

3 A load is pulled along a horizontal straight track, from \(A\) to \(B\), by a force of magnitude \(P \mathrm {~N}\) which acts at an angle of \(30 ^ { \circ }\) upwards from the horizontal. The distance \(A B\) is 80 m . The speed of the load is constant and equal to \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(A\) to the mid-point \(M\) of \(A B\).

1. For the motion from \(A\) to \(M\) the value of \(P\) is 25 . Calculate the work done by the force as the load moves from \(A\) to \(M\). The speed of the load increases from \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(M\) towards \(B\). For the motion from \(M\) to \(B\) the value of \(P\) is 50 and the work done against resistance is the same as that for the motion from \(A\) to \(M\). The mass of the load is 35 kg .
2. Find the gain in kinetic energy of the load as it moves from \(M\) to \(B\) and hence find the speed with which it reaches \(B\).

2 A load of mass 1250 kg is raised by a crane from rest on horizontal ground, to rest at a height of 1.54 m above the ground. The work done against the resistance to motion is 5750 J .

1. Find the work done by the crane.
2. Assuming the power output of the crane is constant and equal to 1.25 kW , find the time taken to raise the load.

7 Loads \(A\) and \(B\), of masses 1.2 kg and 2.0 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical. \(A\) is released and starts to move upwards. It does not reach the pulley in the subsequent motion.

1. Find the acceleration of \(A\) and the tension in the string.
2. Find, for the first 1.5 metres of \(A\) 's motion,
   1. A's gain in potential energy,
   2. the work done on \(A\) by the tension in the string,
   3. A's gain in kinetic energy. B hits the floor 1.6 seconds after \(A\) is released. \(B\) comes to rest without rebounding and the string becomes slack.
   4. Find the time from the instant the string becomes slack until it becomes taut again.