Maximum speed on horizontal road

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 \(B C\).

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 \(A C\).

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the power output of the cyclist.
2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012} \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.

6 A train, of mass 60 tonnes, travels on a straight horizontal track. It has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its engine is working at 800 kW .

1. Find the magnitude of the resistance force acting on the train when the train is travelling at its maximum speed.
2. When the train is travelling at \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power is turned off. Assume that the resistance force is constant and is equal to that found in part (a). Also assume that this resistance force is the only horizontal force acting on the train. Use an energy method to find how far the train travels when slowing from \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   (4 marks)

A car has mass 1200 kg. The maximum power of the car's engine is 32 kW. The resistance to motion due to non-gravitational forces is modelled as a force of constant magnitude 800 N. When the car is travelling on a horizontal road at constant speed \(V\) m s\(^{-1}\), the engine of the car is working at maximum power.

1. Find the value of \(V\). [3]

The car now travels downhill on a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{40}\). The resistance to motion due to non-gravitational forces is still modelled as a force of constant magnitude 800 N. Given that the engine of the car is again working at maximum power,

1. find the acceleration of the car when its speed is 20 m s\(^{-1}\). [4]