Maximum speed on incline vs horizontal

5. A straight road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). A lorry of mass 4800 kg moves up the road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .

1. Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
   (5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
2. the acceleration of the lorry immediately after the road becomes horizontal,
   (3)
3. the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, at which the lorry will go along the horizontal road.
   (3)

4 A car of mass 1200 kg has a maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling on a horizontal road. The car experiences a resistance of \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The maximum power of the car's engine is 45000 W .

1. Show that \(k = 50\).
2. Find the maximum possible acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road.
3. The car climbs a hill, which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal, at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the power of the car's engine.

4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the driving force of the car at this instant.
2. Find the power at this instant. The maximum steady speed of the car on a horizontal road is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle \(\theta ^ { \circ }\) to the horizontal. The maximum steady speed up the slope is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find \(\theta\).

4 A cyclist is riding a bicycle along a straight road which is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. The cyclist is working at a constant rate of 250 W . The combined mass of the cyclist and bicycle is 80 kg and the resistance to their motion is a constant 70 N . Determine the maximum constant speed at which the cyclist can ride the bicycle

• up the hill, and
• down the hill.

1 A car has mass 1200 kg . The total resistance to the car's motion is constant and equal to 250 N .

1. The car is driven along a straight horizontal road with its engine working at 10 kW . Find the acceleration of the car at the instant that its speed is \(5 \mathrm {~ms} ^ { - 1 }\). The maximum power that the car's engine can generate is 20 kW .
2. Find the greatest constant speed at which the car can be driven along a straight horizontal road. The car is driven up a straight road which is inclined at an angle \(\theta\) above the horizontal where \(\sin \theta = 0.05\).
3. Find the greatest constant speed at which the car can be driven up this road.

3 The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW . In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant. At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of the resistance to motion. The truck is being driven along the horizontal road with the engine working at 60 kW .
2. Find the acceleration of the truck at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). The truck is now driven down a straight road which is inclined at an angle \(\theta\) below the horizontal. The greatest constant speed that the truck can be driven at maximum power is \(40 \mathrm {~ms} ^ { - 1 }\).
3. Determine the value of \(\theta\).

2 A car of mass 1100 kg has maximum power of 44000 W . The resistive forces have constant magnitude 1400 N .

1. Calculate the maximum steady speed of the car on the level. The car is moving on a hill of constant inclination \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).
2. Calculate the maximum steady speed of the car when ascending the hill.
3. Calculate the acceleration of the car when it is descending the hill at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) working at half the maximum power.

6. The engine of a car of mass 1200 kg is working at a constant rate of 90 kW as the car moves along a straight horizontal road. The resistive forces opposing the motion of the car are constant and of magnitude 1800 N .

1. Find the acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find, in kJ, the kinetic energy of the car when it is travelling at maximum speed. The car ascends a hill which is straight and makes an angle \(\alpha\) with the horizontal. The power output of the engine and the non-gravitational forces opposing the motion remain the same. Given that the car can attain a maximum speed of \(25 \mathrm {~ms} ^ { - 1 }\),
3. find, in degrees correct to one decimal place, the value of \(\alpha\).
   (5 marks)

3. A car of mass 1200 kg experiences a resistance to motion, \(R\) newtons, which is proportional to its speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the power output of the car engine is 90 kW and the car is travelling along a horizontal road, its maximum speed is \(50 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(R = 36 v\). The car ascends a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 14 }\).
2. Find, correct to 3 significant figures, the maximum speed of the car up the hill assuming that the power output of the engine is unchanged.
   (6 marks)

5. A lorry of mass 40 tonnes moves up a straight road inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 20 }\). The lorry moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In a model of the motion of the lorry, the non-gravitational resistance to motion is assumed to be constant and of magnitude 4400 N .

1. Show that the engine of the lorry is working at a rate of 480 kW . The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion is assumed to remain unchanged.
2. Find the initial acceleration of the lorry.
3. Find, correct to 3 significant figures, the maximum speed of the lorry.
4. Using your answer to part (c), comment on the suitability of the modelling assumption.

1 Brent is riding his bicycle along a straight horizontal road.
While riding along this road Brent can attain a maximum speed of \(6.25 \mathrm {~ms} ^ { - 1 }\) and the wind resistance acting on Brent and his bicycle is constant and equal to 19.2 N . Brent and his bicycle have a combined mass of 72 kg . Brent later begins to ride up a hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal.
Given that the wind resistance and the maximum power developed by the bicycle is unchanged, determine Brent's maximum speed up the hill.

5 A car of mass 1600 kg is travelling uphill along a straight road inclined at \(4.7 ^ { \circ }\) to the horizontal. The power developed by the car is constant and equal to 120 kW . The car is towing a caravan and together they have a maximum speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) uphill. In this question you may model any resistances to motion as negligible.

1. Determine the mass of the caravan. The caravan is now detached from the car. Continuing up the same road, the car passes a point A at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car later passes through a point \(B\) on the same road such that \(A B = 80 \mathrm {~m}\) and the car takes 3.54 seconds to travel from A to B . The power developed by the car while travelling from A to B is constant and equal to 80 kW .
2. Determine the speed of the car at B .
3. State one possible refinement to the model used in parts (a) and (b).

4. A car of mass 1200 kg has an engine that is capable of producing a maximum power of 80 kW . When in motion, the car experiences a constant resistive force of 2000 N .

1. Calculate the maximum possible speed of the car when travelling on a straight horizontal road.
2. The car travels up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). If the car's engine is working at \(80 \%\) capacity, calculate the acceleration of the car at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
3. Explain why the assumption of a constant resistive force may be unrealistic.

7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a total resistive force which can be modelled as being of magnitude \(36 v\) newtons.
The maximum power of the car is 90 kilowatts.
The car starts to descend a hill, inclined at \(5.2 ^ { \circ }\) to the horizontal, along a straight road.
Find the maximum speed of the car down this hill.
[0pt] [5 marks]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{12}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is working at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]
2. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A small car, of mass 850 kg, moves on a straight horizontal road. Its engine is working at its maximum rate of 25 kW, and a constant resisting force of magnitude 900 N opposes the car's motion.

1. Find the acceleration of the car when it is moving with speed 15 ms\(^{-1}\). [3 marks]
2. Find the maximum speed of the car on the horizontal road. [3 marks]

With the engine still working at 25 kW and the non-gravitational resistance remaining at 900 N, the car now climbs a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\).

1. Find the maximum speed of the car on this hill. [4 marks]

An engine of mass \(20\,000\) kg climbs a hill inclined at \(10°\) to the horizontal. The total non-gravitational resistance to its motion has magnitude \(35\,000\) N and the maximum speed of the engine on the hill is \(15\) ms\(^{-1}\).

1. Find, in kW, the maximum rate at which the engine can work. [4 marks]
2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. [3 marks]

Geraint is a cyclist competing in a race along the Taff Trail. The Taff Trail is a track that runs from Cardiff Bay to Brecon. The chart below shows the altitude (height above sea level) along the route. \includegraphics{figure_4} Geraint starts from rest at Cardiff Bay and has a speed of \(10\) ms\(^{-1}\) when he crosses the finish line in Brecon. Geraint and his bike have a total mass of \(80\) kg. The resistance to motion may be modelled by a constant force of magnitude \(16\) N.

1. Given that \(1440\) kJ of energy is used in overcoming resistances during the race,
   1. find the length of the track,
   2. calculate the work done by Geraint. [8]
2. The steepest section of the track may be modelled as a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{2}{7}\). \includegraphics{figure_4b} Geraint is capable of producing a maximum power of \(250\) W. Find the maximum speed that Geraint can attain whilst travelling on this section of the track. [5]