Variable resistance: find k or constants

3 A lorry has mass 12000 kg .

1. The lorry moves at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.08\). At this speed, the magnitude of the resistance to motion on the lorry is 1500 N . Show that the power of the lorry's engine is 55.5 kW .
   When the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the magnitude of the resistance to motion is \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant.
2. Show that \(k = 60\).
3. The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at 55.5 kW , find the lorry's speed.

4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate at which the cyclist is working.
   (6 marks)

2 A car of mass 1250 kg travels along a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The car travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope and the engine of the car works at a constant rate of 21 kW .

1. Calculate the value of \(k\).
2. Calculate the constant speed of the car on a horizontal road.

1 Throughout all parts of this question, the resistance to the motion of a car has magnitude \(\mathrm { kv } ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. At first, the car travels along a straight horizontal road with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power developed by the car at this speed is 5000 W .

1. Show that \(k = \frac { 5 } { 8 }\).
2. Find the power the car must develop in order to maintain a constant speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling along the same horizontal road. The car climbs a hill which is inclined at an angle of \(2 ^ { \circ }\) to the horizontal. The power developed by the car is 13000 W , and the car has a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Determine the mass of the car.

1. A car of mass 1000 kg moves along a straight horizontal road.

In all circumstances, when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(c v ^ { 2 } \mathrm {~N}\), where \(c\) is a constant. The maximum power that can be developed by the engine of the car is 50 kW .
At the instant when the speed of the car is \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the engine is working at its maximum power, the acceleration of the car is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Convert \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) into \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the acceleration of the car at the instant when the speed of the car is \(144 \mathrm { kmh } ^ { - 1 }\) and the engine is working at its maximum power. The maximum speed of the car when the engine is working at its maximum power is \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
3. Find, to the nearest whole number, the value of \(V\).

1. The total mass of a cyclist and his bicycle is 100 kg .

In all circumstances, the magnitude of the resistance to the motion of the cyclist from non-gravitational forces is modelled as being \(k v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The cyclist can freewheel, without pedalling, down a slope that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 35 }\), at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When he is pedalling up a slope that is inclined to the horizontal at an angle \(\beta\), where \(\sin \beta = \frac { 1 } { 70 }\), and he is moving at the same constant speed \(V \mathrm {~ms} ^ { - 1 }\), he is working at a constant rate of \(P\) watts.

1. Find \(P\) in terms of \(V\). If he pedals and works at a rate of 35 V watts on a horizontal road, he moves at a constant speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find \(U\) in terms of \(V\).

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

A car has mass 1400 kg. When the speed of the car is \(v\text{ ms}^{-1}\) the magnitude of the resistance to motion is \(kv^2\) N where \(k\) is a constant.

1. The car moves at a constant speed of \(24\text{ ms}^{-1}\) up a hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.12\). At this speed the magnitude of the resistance to motion is 480 N.
   1. Find the value of \(k\). [1]
   2. Find the power of the car's engine. [3]
2. The car now moves at a constant speed on a straight level road. Given that its engine is working at 54 kW, find this speed. [3]

A motor-cycle and its rider have a total mass of 460 kg. The maximum rate at which the cycle's engine can work is 25 920 W and the maximum speed of the cycle on a horizontal road is 36 ms\(^{-1}\). A variable resisting force acts on the cycle and has magnitude \(kv^2\), where \(v\) is the speed of the cycle in ms\(^{-1}\).

1. Show that \(k = \frac{5}{8}\). [4 marks]
2. Find the acceleration of the cycle when it is moving at 25 ms\(^{-1}\) on the horizontal road, with its engine working at full power. [5 marks]

A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(kv^{\frac{3}{2}}\) N, where \(v\) ms\(^{-1}\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W, the car has speed 15 ms\(^{-1}\) and is accelerating at 0.4 ms\(^{-2}\).

1. Find the value of \(k\). [4]

It is given that the greatest steady speed of the car on this road is 30 ms\(^{-1}\).

1. Find the greatest power that the engine can produce. [3]

Kang is riding a motorbike along a straight, horizontal road. The motorbike has a maximum power of 75 000 W The maximum speed of the motorbike is \(50 \text{ m s}^{-1}\) When the speed of the motorbike is \(v \text{ m s}^{-1}\), the resistance force is \(kv\) newtons. Find the value of \(k\) Fully justify your answer. [4 marks]

A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).

1. Show that \(\lambda = 8\). [4]
2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), 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. Find the acceleration of the car immediately after the towbar breaks. [4]
3. Use the work-energy principle to find the value of \(d\). [4]