6.02l Power and velocity: P = Fv

2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).

5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the resistance to the cyclist's motion is 16 N .
2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).

2. A car of mass 500 kg is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 150 N .

1. Find the rate of working of the engine of the car. When the car is travelling up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car then comes to instantaneous rest, without braking, having moved a distance \(d\) metres up the road from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 150 N .
2. Use the work-energy principle to find the value of \(d\).

6. A car of mass 800 kg pulls a trailer of mass 300 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 14 }\). The trailer is attached to the car by a light inextensible towbar which is parallel to the direction of motion of the car. The car's engine works at a constant rate of \(P \mathrm {~kW}\). The non-gravitational resistances to motion are constant and of magnitude 600 N on the car and 200 N on the trailer. At a given instant the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the value of \(P\). When the car is moving up the road at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. The non-gravitational resistance to the motion of the trailer remains constant and of magnitude 200 N .
2. Find, using the work-energy principle, the value of \(d\).
   

1. A car of mass 900 kg is moving on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 49 }\). When the car is moving up the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.

When the car is moving down the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is still modelled as a constant force of magnitude \(R\) newtons. Find

1. the value of \(R\),
2. the value of \(v\).

8. \begin{figure}[h] \end{figure} A rough ramp \(A B\) is fixed to horizontal ground at \(A\). The ramp is inclined at \(20 ^ { \circ }\) to the ground. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 6 \mathrm {~m}\). The point \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 3 kg is projected with speed \(15 \mathrm {~ms} ^ { - 1 }\) from \(A\) towards \(B\). At the instant \(P\) reaches the point \(B\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force due to friction is modelled as a constant force of magnitude \(F\) newtons.

1. Use the work-energy principle to find the value of \(F\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). The speed of \(P\) as it hits the ground at \(C\) is \(w \mathrm {~ms} ^ { - 1 }\). Find
2. 1. the value of \(w\),
   2. the direction of motion of \(P\) as it hits the ground at \(C\),
3. the greatest height of \(P\) above the ground as \(P\) moves from \(A\) to \(C\).

1. A cyclist and his bicycle have a total mass of 75 kg . The cyclist is moving down a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\)

The cyclist is working at a constant rate of 56 W . The magnitude of the resistance to motion is modelled as a constant force of magnitude 40 N . At the instant when the speed of the cyclist is \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\), his acceleration is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 2 }\) Find the value of \(V\).
(5)

3. A car of mass 600 kg travels along a straight horizontal road with the engine of the car working at a constant rate of \(P\) watts. The resistance to the motion of the car is modelled as a constant force of magnitude \(R\) newtons. At the instant when the speed of the car is \(15 \mathrm {~ms} ^ { - 1 }\), the magnitude of the acceleration of the car is \(0.2 \mathrm {~ms} ^ { - 2 }\). Later the same car travels up a straight road inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the engine of the car is working at a constant rate of \(P\) watts, the car has a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). Find the value of \(P\).

2. A car of mass 600 kg tows a trailer of mass 200 kg up a hill along a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The trailer is attached to the car by a light inextensible towbar. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude 150 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 300 N . When the engine of the car is working at a constant rate of \(P \mathrm {~kW}\) the car and the trailer have a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the value of \(P\). Later, at the instant when the car and the trailer are travelling up the hill with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. When the towbar breaks the trailer is at the point \(X\). The trailer continues to travel up the hill before coming to instantaneous rest at the point \(Y\). The resistance to the motion of the trailer from non-gravitational forces is again modelled as a constant force of magnitude 300 N .
2. Use the work-energy principle to find the distance \(X Y\).
   VIIV SIHI NI III M I0N 00 :

1. A truck of mass 1500 kg is moving on a straight horizontal road.

The engine of the truck is working at a constant rate of 30 kW .
The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the truck is moving at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the value of \(R\). Later on, the truck is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 8 }\) The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 500 N .
   The engine of the truck is again working at a constant rate of 30 kW . At the instant when the speed of the truck is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(V\)

5. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 4. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 250 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 150 N . The towbar is modelled as a light rod.
At the instant when the speed of the van is \(16 \mathrm {~ms} ^ { - 1 }\), the engine of the van is working at a rate of 10 kW .

1. Find the deceleration of the van at this instant.
2. Find the tension in the towbar at this instant.

2. A trailer of mass 250 kg is towed by a car of mass 1000 kg . The car and the trailer are travelling down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) The resistance to motion of the car is modelled as a single force of magnitude 300 N acting parallel to the road. The resistance to motion of the trailer is modelled as a single force of magnitude 100 N acting parallel to the road. The towbar joining the car to the trailer is modelled as a light rod which is parallel to the direction of motion. At a given instant the car and the trailer are moving down the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the power being developed by the car's engine at this instant.
2. Find the tension in the towbar at this instant.

2. A truck of mass 1800 kg is moving along a straight horizontal road. The engine of the truck is working at a constant rate of 10 kW . The non-gravitational resistance to motion is modelled as a constant force of magnitude \(R\) newtons. At the instant when the truck is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\).
   (4) The truck now moves up a straight road at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The non-gravitational resistance to motion is now modelled as a constant force of magnitude 30 V newtons. The engine of the truck is now working at a constant rate of 12 kW .
2. Find the value of \(V\).
   DO NOT WIRITE IN THIS AREA
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{790546bf-38a4-4eb7-876e-941fe58f4a48-05_529_1040_118_450} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}

3. A cyclist and his bicycle, with a combined mass of 75 kg , move along a straight horizontal road. The cyclist is working at a constant rate of 180 W . There is a constant resistance to the motion of the cyclist and his bicycle of magnitude \(R\) newtons. At the instant when the speed of the cyclist is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). Later, the cyclist moves up a straight road with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 21 }\). The cyclist is working at a rate of 180 W and the resistance to the motion of the cyclist and his bicycle from non-gravitational forces is again the same constant force of magnitude \(R\) newtons.
2. Find the value of \(v\).

1. A truck of weight 9000 N is travelling up a hill on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\)

When the truck travels up the hill with the engine working at \(3 P\) watts, the truck is moving at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Later on, the truck travels down the hill along the same road, with the engine working at \(P\) watts. At the instant when the speed of the truck is \(12 \mathrm {~ms} ^ { - 1 }\), the acceleration of the truck is \(\frac { g } { 20 }\) The resistance to motion of the truck from non-gravitational forces is a constant force of magnitude \(R\) newtons in all circumstances. Find (i) the value of \(P\),
(ii) the value of \(R\).
WIHW SIHI NIT INHM ION OC
WIIV SIHI NI III IM I ON OC
VAYV SIHI NI JLIUM ION OO

1. A motorcyclist and his motorcycle have a combined mass of 480 kg .

The motorcyclist drives down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 12 }\), with the engine of the motorcycle working at 3.5 kW . The motorcycle is moving at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the motorcycle is modelled as a constant force with magnitude 20 V newtons. Find the value of \(V\).
(5)

4. A truck of mass 900 kg is moving along a straight horizontal road with the engine of the truck working at a constant rate of \(P\) watts. The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the speed of the truck is \(15 \mathrm {~ms} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Later the same truck is moving down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to the motion of the truck is again modelled as a constant force of magnitude \(R\) newtons. The engine of the truck is again working at a constant rate of \(P\) watts.
At the instant when the speed of the truck is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find the value of \(R\).

1. A van of mass 900 kg is moving along a straight horizontal road.

The resistance to the motion of the van is modelled as a constant force of magnitude 600 N . The engine of the van is working at a constant rate of 24 kW .
At the instant when the speed of the van is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(2 \mathrm {~ms} ^ { - 2 }\)

1. Find the value of \(V\) Later on, the van is towing a trailer of mass 700 kg up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) The trailer is attached to the van by a towbar, as shown in Figure 3.
   The towbar is parallel to the direction of motion of the van and the trailer. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-20_367_1194_1091_438} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 600 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 550 N . The towbar is modelled as a light rod.
   The engine of the van is working at a constant rate of 24 kW .
2. Find the acceleration of the van at the instant when the van and the trailer are moving with speed \(8 \mathrm {~ms} ^ { - 1 }\) At the instant when the van and the trailer are moving up the road at \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. The trailer continues to move in a straight line up the road until it comes to instantaneous rest. The distance moved by the trailer as it slows from a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to instantaneous rest is \(d\) metres.
3. Use the work-energy principle to find the value of \(d\).

2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .

1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find, in kW , the rate at which the van's engine is now working.
   "

2. \begin{figure}[h] \end{figure} A truck of mass 1200 kg is being driven up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The resistance to the motion of the truck from non-gravitational forces is modelled as a single constant force of magnitude 250 N . Two points, \(A\) and \(B\), lie on the road, with \(A B = 90 \mathrm {~m}\). The speed of the truck at \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of the truck at \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 2. The truck is modelled as a particle and the road is modelled as a straight line.

1. Find the work done by the engine of the truck as the truck moves from \(A\) to \(B\). On another occasion, the truck is being driven down the same road. The resistance to the motion of the truck is modelled as a single constant force of magnitude 250 N . The engine of the truck is working at a constant rate of 8 kW .
2. Find the acceleration of the truck at the instant when its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. A vehicle of mass 450 kg is moving on a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\) At the instant when the vehicle is moving down the road at \(12 \mathrm {~ms} ^ { - 1 }\)

• the engine of the vehicle is working at a rate of \(P\) watts
• the acceleration of the vehicle is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
• the resistance to the motion of the vehicle is modelled as a constant force of magnitude \(R\) newtons

At the instant when the vehicle is moving up the road at \(12 \mathrm {~ms} ^ { - 1 }\)

• the engine of the vehicle is working at a rate of \(2 P\) watts
• the deceleration of the vehicle is \(0.5 \mathrm {~ms} ^ { - 2 }\)
• the resistance to the motion of the vehicle from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons

Find the value of \(P\).

2. A car of mass 900 kg is moving down a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The engine of the car is working at a constant rate of 15 kW .
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N . Find the acceleration of the car at the instant when it is moving at \(16 \mathrm {~ms} ^ { - 1 }\) \section*{Qu}

1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .

The total mass of the cyclist and her cycle is 80 kg .
The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .

1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
   The distance from \(A\) to \(B\) is 20 km .
   Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
   The cyclist starts from rest at \(A\).
   At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
3. Using this model, find the value of \(P\)