6.02l Power and velocity: P = Fv

3 A car of mass 1250 kg travels along a horizontal straight road with increasing speed. The power provided by the car's engine is constant and equal to 24 kW . The resistance to the car's motion is constant and equal to 600 N .

1. Show that the speed of the car cannot exceed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car at an instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of 40 kW . The resistance to motion of the car is constant and equal to 800 N . The car passes through the point \(A\) on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).

2 A cyclist, working at a constant rate of 400 W , travels along a straight road which is inclined at \(2 ^ { \circ }\) to the horizontal. The total mass of the cyclist and his cycle is 80 kg . Ignoring any resistance to motion, find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling

1. uphill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. downhill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 A car of mass 1250 kg travels along a horizontal straight road. The power of the car's engine is constant and equal to 24 kW and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). The car passes through the point \(A\) on the road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). The car continues with increasing speed, passing through the point \(B\) on the road with speed \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car subsequently passes through the point \(C\).
2. Find the acceleration of the car at \(B\), giving the answer in \(\mathrm { m } \mathrm { s } ^ { - 2 }\) correct to 3 decimal places.
3. Show that, while the car's speed is increasing, it cannot reach \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Explain why the speed of the car is approximately constant between \(B\) and \(C\).
5. State a value of the approximately constant speed, and the maximum possible error in this value at any point between \(B\) and \(C\). The work done by the car's engine during the motion from \(B\) to \(C\) is 1200 kJ .
6. By assuming the speed of the car is constant from \(B\) to \(C\), find, in either order,
   1. the approximate time taken for the car to travel from \(B\) to \(C\),
   2. an approximation for the distance \(B C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-3_218_1280_1146_431} \(A B\) and \(B C\) are straight roads inclined at \(5 ^ { \circ }\) to the horizontal and \(1 ^ { \circ }\) to the horizontal respectively. \(A\) and \(C\) are at the same horizontal level and \(B\) is 45 m above the level of \(A\) and \(C\) (see diagram, which is not to scale). A car of mass 1200 kg travels from \(A\) to \(C\) passing through \(B\).

1. For the motion from \(A\) to \(B\), the speed of the car is constant and the work done against the resistance to motion is 360 kJ . Find the work done by the car's engine from \(A\) to \(B\). The resistance to motion is constant throughout the whole journey.
2. For the motion from \(B\) to \(C\) the work done by the driving force is 1660 kJ . Given that the speed of the car at \(B\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that its speed at \(C\) is \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. The car's driving force immediately after leaving \(B\) is 1.5 times the driving force immediately before reaching \(C\). Find, correct to 2 significant figures, the ratio of the power developed by the car's engine immediately after leaving \(B\) to the power developed immediately before reaching \(C\).

1 A racing cyclist, whose mass with his cycle is 75 kg , works at a rate of 720 W while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to \(R \mathrm {~N}\).

1. Given that the cyclist is accelerating at \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when his speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(R\).
2. Given that the cyclist's acceleration is positive, show that his speed is less than \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 A car of mass 600 kg travels along a straight horizontal road starting from a point \(A\). The resistance to motion of the car is 750 N .

1. The car travels from \(A\) to \(B\) at constant speed in 100 s . The power supplied by the car's engine is constant and equal to 30 kW . Find the distance \(A B\).
2. The car's engine is switched off at \(B\) and the car's speed decreases until the car reaches \(C\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(B C\).
3. The car's engine is switched on at \(C\) and the power it supplies is constant and equal to 30 kW . The car takes 14 s to travel from \(C\) to \(D\) and reaches \(D\) with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(C D\).

7 A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of the car is constant and has magnitude 960 N . The car's engine works at a rate of 17280 W .

1. Calculate the acceleration of the car at an instant when its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car passes through the points \(A\) and \(B\). While the car is moving between \(A\) and \(B\) it has constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that \(V = 18\). At the instant that the car reaches \(B\) the engine is switched off and subsequently provides no energy. The car continues along the straight line until it comes to rest at the point \(C\). The time taken for the car to travel from \(A\) to \(C\) is 52.5 s .
3. Find the distance \(A C\).

5 A lorry of mass 15000 kg climbs from the bottom to the top of a straight hill, of length 1440 m , at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N .

1. Find the work done by the driving force. On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point \(P\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and is now equal to 1600 N . The work done by the lorry's engine from the top of the hill to the point \(P\) is 5030 kJ .
2. Find the distance from the top of the hill to the point \(P\).

3 The resistance to motion acting on a runner of mass 70 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the runner's speed and \(k\) is a constant. The greatest power the runner can exert is 100 W . The runner's greatest steady speed on horizontal ground is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 6.25\).
2. Find the greatest steady speed of the runner while running uphill on a straight path inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).

6 A lorry of mass 12500 kg travels along a road from \(A\) to \(C\) passing through a point \(B\). The resistance to motion of the lorry is 4800 N for the whole journey from \(A\) to \(C\).

1. The section \(A B\) of the road is straight and horizontal. On this section of the road the power of the lorry's engine is constant and equal to 144 kW . The speed of the lorry at \(A\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration at \(B\) is \(0.096 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the acceleration of the lorry at \(A\) and show that its speed at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. The section \(B C\) of the road has length 500 m , is straight and inclined upwards towards \(C\). On this section of the road the lorry's driving force is constant and equal to 5800 N . The speed of the lorry at \(C\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the height of \(C\) above the level of \(A B\).

1 A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of 22.5 kW . Find the resistance to the car's motion at an instant when the car's speed is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3 A train of mass 200000 kg moves on a horizontal straight track. It passes through a point \(A\) with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and later it passes through a point \(B\). The power of the train's engine at \(B\) is 1.2 times the power of the train's engine at \(A\). The driving force of the train's engine at \(B\) is 0.96 times the driving force of the train's engine at \(A\).

1. Show that the speed of the train at \(B\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. For the motion from \(A\) to \(B\), find the work done by the train's engine given that the work done against the resistance to the train's motion is \(2.3 \times 10 ^ { 6 } \mathrm {~J}\).

3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .

1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .

7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.

1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
3. the distance \(A B\). {www.cie.org.uk} after the live examination series. }

5 A cyclist and his bicycle have a total mass of 90 kg . The cyclist starts to move with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a straight hill, of length 500 m , which is inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, \(R \mathrm {~N}\), and the cyclist's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), both vary.

1. Show that \(R = \frac { 420 } { v } + 43.56\).
2. Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of \(R\) when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill.

7 A straight hill \(A B\) has length 400 m with \(A\) at the top and \(B\) at the bottom and is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. A straight horizontal road \(B C\) has length 750 m . A car of mass 1250 kg has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) when starting to move down the hill. While moving down the hill the resistance to the motion of the car is 2000 N and the driving force is constant. The speed of the car on reaching \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By using work and energy, find the driving force of the car. On reaching \(B\) the car moves along the road \(B C\). The driving force is constant and twice that when the car was on the hill. The resistance to the motion of the car continues to be 2000 N . Find
2. the acceleration of the car while moving from \(B\) to \(C\),
3. the power of the car's engine as the car reaches \(C\).

6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at \(10 ^ { \circ }\) above the horizontal. The block starts from rest and travels a distance of 20 m . There is a constant resistance force of magnitude 30 N opposing motion.

1. Find the work done by the pulling force.
2. Use an energy method to find the speed of the block when it has moved a distance of 20 m .
3. Find the greatest power exerted by the 50 N force. \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_236_1027_2161_566} After the block has travelled the 20 m , it comes to a plane inclined at \(5 ^ { \circ }\) to the horizontal. The force of 50 N is now inclined at an angle of \(10 ^ { \circ }\) to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N .
4. Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
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6 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the van and the trailer is modelled as two particles connected by a light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on the trailer.

1. Find the power of the van's engine.
2. Write down the tension in the cable. The van reaches the bottom of a hill inclined at \(4 ^ { \circ }\) to the horizontal with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the van's engine is increased to 25000 W .
3. Assuming that the resistance forces remain the same, find the new tension in the cable at the instant when the speed of the van up the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m . There is a constant resistance to motion of 25 N .

1. Find the work done by the crane.
2. Given that the time taken to raise the block is 2 s , find the power of the crane.

6 A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant resistance to motion of 20 N . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the total mass of the cyclist and bicycle is 80 kg . The cyclist comes to a hill inclined at \(2 ^ { \circ }\) to the horizontal. When the cyclist starts climbing the hill, he increases his power to a constant 300 W . The resistance to motion remains 20 N .
2. Show that the steady speed up the hill which the cyclist can maintain when working at this power is \(6.26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the acceleration at an instant when the cyclist is travelling at \(90 \%\) of the speed in part (ii).

2 A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to motion is 1150 N .

1. Find the power output of the tractor's engine.
   The tractor comes to a hill inclined at \(4 ^ { \circ }\) above the horizontal. The power output is increased to 25 kW and the resistance to motion is unchanged.
2. Find the deceleration of the tractor at the instant it begins to climb the hill.
3. Find the constant speed that the tractor could maintain on the hill when working at this power.

2 A lorry of mass 7850 kg travels on a straight hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 1480 N .

1. Find the power of the lorry's engine when the lorry is going up the hill at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the power of the lorry's engine at an instant when the lorry is going down the hill at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

2 A high-speed train of mass 490000 kg is moving along a straight horizontal track at a constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engines are supplying 4080 kW of power.

1. Show that the resistance force is 48000 N .
2. The train comes to a hill inclined at an angle \(\theta ^ { \circ }\) above the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 200 }\). Given that the resistance force is unchanged, find the power required for the train to keep moving at the same constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.

1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
   When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.