6.02e Calculate KE and PE: using formulae

5 A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the motion of the car and caravan are 440 N and 280 N respectively. The car and caravan are connected by a light rigid tow-bar.

1. The car and caravan move along a horizontal part of the road at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate, in kW , the power developed by the engine of the car.
   2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car and caravan and the tension in the tow-bar.
2. The car and caravan now travel along a part of the road inclined at \(\sin ^ { - 1 } 0.06\) to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at 28 kW .
   1. Find this constant speed.
   2. Find the increase in the potential energy of the caravan in one minute.

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 \(C\).

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 box of mass 5 kg is pulled at a constant speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) for 15 s 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 change in gravitational potential energy of the box.
2. Find the work done by the pulling force.

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 \(B\).

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_502_661_1219_742} A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point \(O\) on the ground, using a winding drum. The load passes through a point \(A , 20 \mathrm {~m}\) above \(O\), with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Find, for the motion from \(O\) to \(A\),

1. the gain in the potential energy of the load,
2. the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.
3. Given that the work done against the resistance to motion from \(O\) to \(A\) is 20 kJ and that the time taken for the load to travel from \(O\) to \(A\) is 41.7 s , find the power output of the winding drum.

7 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_168_803_1909_671} The frictional force acting on a small block of mass 0.15 kg , while it is moving on a horizontal surface, has magnitude 0.12 N . The block is set in motion from a point \(X\) on the surface, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits a vertical wall at a point \(Y\) on the surface 2 s later. The block rebounds from the wall and moves directly towards \(X\) before coming to rest at the point \(Z\) (see diagram). At the instant that the block hits the wall it loses 0.072 J of its kinetic energy. The velocity of the block, in the direction from \(X\) to \(Y\), is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after it leaves \(X\).

1. Find the values of \(v\) when the block arrives at \(Y\) and when it leaves \(Y\), and find also the value of \(t\) when the block comes to rest at \(Z\). Sketch the velocity-time graph.
2. The displacement of the block from \(X\), in the direction from \(X\) to \(Y\), is \(s \mathrm {~m}\) at time \(t \mathrm {~s}\). Sketch the displacement-time graph. Show on your graph the values of \(s\) and \(t\) when the block is at \(Y\) and when it comes to rest at \(Z\).

2 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle \(A\) of mass 1.6 kg and particle \(B\) of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle \(\theta\), where \(\sin \theta = 0.8\). Particle \(A\) is held at rest at the bottom of the plane and \(B\) hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). \(A\) is released from rest and the particles start to move.

1. Show that the loss of potential energy of the system, when \(B\) reaches the level of the bottom of the plane, is 23.328 J .
2. Hence find the speed of the particles when \(B\) reaches the level of the bottom of the plane.

4 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_227_586_1631_781} The diagram shows a vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section. A particle of mass 0.15 kg is released from rest at \(A\).

1. Assuming that the particle reaches \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there are no resistances to motion, find the height of \(A\) above \(B\).
2. Assuming instead that the particle reaches \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and that the height of \(A\) above \(B\) is 4 m , find the work done against the resistances to motion.

4 A lorry of mass 16000 kg climbs from the bottom to the top of a straight hill of length 1000 m at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 20 m above the level of the bottom of the hill. The driving force of the lorry is constant and equal to 5000 N . Find

1. the gain in gravitational potential energy of the lorry,
2. the work done by the driving force,
3. the work done against the force resisting the motion of the lorry. On reaching the top of the hill the lorry continues along a straight horizontal road against a constant resistance of 1500 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(P\). The distance of \(P\) from the top of the hill is 2000 m .
4. Find the work done by the driving force of the lorry while the lorry travels from the top of the hill to \(P\).

2 A crate of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle \(\alpha ^ { \circ }\) upwards from the horizontal. The total resistance to motion of the crate has constant magnitude 250 N . The crate starts from rest at the point \(O\) and passes the point \(P\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O P\) is 20 m . For the crate's motion from \(O\) to \(P\), find

1. the increase in kinetic energy of the crate,
2. the work done against the resistance to the motion of the crate,
3. the value of \(\alpha\).

1 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_421_1223_267_461} A box of mass 8 kg is pulled, at constant speed, up a straight path which is inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The pulling force is constant, of magnitude 30 N , and acts upwards at an angle of \(10 ^ { \circ }\) from the path (see diagram). The box passes through the points \(A\) and \(B\), where \(A B = 20 \mathrm {~m}\) and \(B\) is above the level of \(A\). For the motion from \(A\) to \(B\), find

1. the work done by the pulling force,
2. the gain in potential energy of the box,
3. the work done against the resistance to motion of the box.

5 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-3_515_789_995_676} The diagram shows the vertical cross-section \(L M N\) of a fixed smooth surface. \(M\) is the lowest point of the cross-section. \(L\) is 2.45 m above the level of \(M\), and \(N\) is 1.2 m above the level of \(M\). A particle of mass 0.5 kg is released from rest at \(L\) and moves on the surface until it leaves it at \(N\). Find

1. the greatest speed of the particle,
2. the kinetic energy of the particle at \(N\). The particle is now projected from \(N\), with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface towards \(M\).
3. Find the least value of \(v\) for which the particle will reach \(L\).

7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.

1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
3. Find the speed with which the particle returns to \(P\).