Towing system: horizontal road

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.

6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

1. The car and caravan are travelling along a straight horizontal road.
   1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
   2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).

6 A car of mass 1750 kg is pulling a caravan of mass 500 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 650 N and 150 N respectively.

1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the power of the car's engine.
   2. The engine's power is now suddenly increased to 40 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).

5 The engine of a car of mass 1200 kg produces a maximum power of 40 kW .
In an initial model of the motion of the car the total resistance to motion is assumed to be constant.

1. Given that the greatest steady speed of the car on a straight horizontal road is \(42 \mathrm {~ms} ^ { - 1 }\), find the magnitude of the resistance force. The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
2. Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the power of the engine of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
4. Without carrying out any further calculations,
   (a) explain whether the time taken to attain a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
   (b) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

2 A car of mass 1350 kg travels along a straight horizontal road. Throughout this question the resistance force to the motion of the car is modelled as constant and equal to 920 N .

1. Calculate the power, in kW , developed by the car when the car is travelling at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car is now used to tow a caravan of mass 1050 kg along the same road. When the car tows the caravan at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) the power developed by the car is 45 kW .
2. Find the additional resistance force due to the caravan. In the remaining parts of this question the power developed by the car is constant and equal to 68 kW and the resistance force due to the caravan is modelled as constant and equal to the value found in part (ii). When the car and caravan pass a point A on the same straight horizontal road the speed of the car and caravan is \(20 \mathrm {~ms} ^ { - 1 }\).
3. Find the acceleration of the car and caravan at point A . The car and caravan later pass a point B on the same straight horizontal road with speed \(28 \mathrm {~ms} ^ { - 1 }\). The distance \(A B\) is \(1024 m\).
4. Find the time taken for the car and caravan to travel from point A to point B .
5. Suggest one way in which any of the modelling assumptions used in this question could have been improved.

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
   The tow rope is modelled as being light and inextensible.
   Using the model,
2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)

4 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-3_216_1219_255_415} \(A\) and \(B\) are two long straight parallel horizontal sections of railway track. An engine on track \(A\) is attached to a carriage of mass 6000 kg on track \(B\) by a light inextensible chain which remains horizontal and taut in the ensuing motion. The chain is 13 m in length and the points of attachment on the engine and carriage are a perpendicular distance of 5 m apart. The engine and carriage start at rest and then the engine accelerates uniformly to a speed of \(5.6 \mathrm {~ms} ^ { - 1 }\) while travelling 250 m . It is assumed that any resistance to motion can be ignored.

1. Find the work done on the carriage by the tension in the chain.
2. Find the magnitude of the tension in the chain. The mass of the engine is 10000 kg .
3. At a point further along the track the engine and the carriage are moving at a speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) and the power of the engine is 68 kW . Find the acceleration of the engine at this instant.

4 A car has a mass of 850 kg and its engine can generate a maximum power of 35 kW . The total resistance to motion of the car is modelled as \(k v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. When the car is moving in a straight line on a straight horizontal road, the greatest constant speed that it can attain is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 56\).
2. Find the greatest possible acceleration of the car on the road at an instant when it is moving with a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A trailer of mass 240 kg is attached to the car by means of a light inextensible tow bar which is parallel to the surface of the road. The resistance to motion of the trailer is modelled as a constant force of magnitude 350 N . The car and trailer move on the horizontal road. At a certain instant the car's engine is working at a rate of 30 kW and the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the speed of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. The car and trailer now move in a straight line on a straight road inclined at \(8 ^ { \circ }\) to the horizontal.
4. Find the difference between their greatest possible constant speed travelling up the slope and their greatest possible constant speed travelling down the slope.

2 A car of mass 800 kg is driven with its engine generating a power of 15 kW .

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude 150 N . Given that \(\sin \theta = \frac { 1 } { 20 }\), find the speed of the car.
3. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming that the resistance to motion of the car is three times the resistance to motion of the trailer, find

1 A car of mass 800 kg is driven with its engine generating a power of 15 kW .

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude 150 N . Given that \(\sin \theta = \frac { 1 } { 20 }\), find the speed of the car.
3. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming that the resistance to motion of the car is three times the resistance to motion of the trailer, find

A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of \(2\,\text{m}\,\text{s}^{-1}\). There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.

1. State the tension in the coupling. [1]
2. Find the power produced by the locomotive's engine. [1] The power produced by the locomotive's engine is now changed to 1.2 W.
3. Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate. [5]

A car of mass 800 kg is driven with its engine generating a power of 15 kW.

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
   1. the resistance to motion of the car,
   2. the magnitude of the tension in the towbar
   [4]