6.02l Power and velocity: P = Fv

A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5°\) to the horizontal. At a certain point \(P\) on the hill the car's speed is 20 m s\(^{-1}\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is 15 m s\(^{-1}\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. [4]

Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.

1. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW, correct to 3 significant figures. [3]
2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]

A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find

1. the work done by the force, [3]
2. the power with which the force is working. [2]

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]

The maximum power produced by the engine of a small aeroplane of mass 2 tonnes is 128 kW. Air resistance opposes the motion directly and the lift force is perpendicular to the direction of motion. The magnitude of the air resistance is proportional to the square of the speed and the maximum steady speed in level flight is \(80 \text{ ms}^{-1}\).

1. Calculate the magnitude of the air resistance when the speed is \(60 \text{ ms}^{-1}\). [5]

The aeroplane is climbing at a constant angle of \(2°\) to the horizontal.

1. Find the maximum acceleration at an instant when the speed of the aeroplane is \(60 \text{ ms}^{-1}\). [4]

A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N. At a certain instant the car is accelerating at \(0.3 \text{ m s}^{-2}\) and the engine of the car is working at a rate of 23 kW.

1. Find the speed of the car at this instant. [3]

Subsequently the car moves up a hill inclined at \(10°\) to the horizontal at a steady speed of \(12 \text{ m s}^{-1}\). The resistance to motion is still a constant 600 N.

1. Calculate the power of the car's engine as it moves up the hill. [3]

A cyclist and her bicycle have a combined mass of 80 kg.

1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
   1. How much energy is gained by the cyclist and bicycle? [2]
   2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
   1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
   2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]

A motor-cycle, whose mass including the rider is \(120\) kg, is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \text{ m s}^{-1}\) and when it has travelled a distance of \(x\) m beyond \(A\) its speed is \(v \text{ m s}^{-1}\). The engine develops a constant power of \(8\) kW and resistances are modelled by a force of \(0.25v^2\) N opposing the motion.

1. Show that \(\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1\). [5]
2. Find the speed of the motor-cycle when it has travelled \(500\) m beyond \(A\). [6]

A car of mass \(M\) moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude \(R\). The engine of the car works at a constant rate \(RU\). Find the time taken for the car to accelerate from a speed of \(\frac{1}{4}U\) to a speed of \(\frac{1}{2}U\). [9]

A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(kv^2\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),

1. show that \(V = \sqrt{\frac{F}{k}}\). [2]

Given instead that the lorry starts from rest,

1. show that the distance travelled by the lorry in attaining a speed of \(\frac{1}{2}V\) is $$\frac{M}{2k}\ln\left(\frac{4}{3}\right).$$ [9]

A train of mass \(m\) is moving along a straight horizontal railway line. A time \(t\), the train is moving with speed \(v\) and the resistance to motion has magnitude \(kv\), where \(k\) is a constant. The engine of the train is working at a constant rate \(P\).

1. Show that, when \(v > 0\), \(mv\frac{dv}{dt} + kv^2 = P\). [3]

When \(t = 0\), the speed of the train is \(\frac{1}{3}\sqrt{\frac{P}{k}}\).

1. Find, in terms of \(m\) and \(k\), the time taken for the train to double its initial speed. [8]

A lorry of mass \(M\) moves along a straight horizontal road against a constant resistance of magnitude \(R\). The engine of the lorry works at a constant rate \(RU\), where \(U\) is a constant. At time \(t\), the lorry is moving with speed \(v\).

1. Show that \(Mv\frac{dv}{dt} = R(U - v)\). [3]

At time \(t = 0\), the lorry has speed \(\frac{1}{4}U\) and the time taken by the lorry to attain a speed of \(\frac{3}{4}U\) is \(\frac{kMU}{R}\), where \(k\) is a constant.

1. Find the exact value of \(k\). [7]

A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW. When the speed of the car is \(v\) m s\(^{-1}\), the resistance to motion has magnitude \(10v\) newtons.

1. Show that, at the instant when \(v = 20\), the acceleration of the car is 1.05 m s\(^{-2}\). [3]
2. Find the distance travelled by the car as it accelerates from a speed of 10 m s\(^{-1}\) to a speed of 20 m s\(^{-1}\). [8]

A car of mass 1000 kg has a maximum speed of \(40\,\text{m}\,\text{s}^{-1}\) when travelling on a straight horizontal race track. The maximum power output of the car's engine is 48 kW The total resistance force experienced by the car can be modelled as being proportional to the car's speed. Find the maximum possible acceleration of the car when it is travelling at \(25\,\text{m}\,\text{s}^{-1}\) on the straight horizontal race track. Fully justify your answer. [7 marks]

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 1250 kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v \, \text{m s}^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.54 \, \text{m s}^{-2}\). At a point \(B\) on the road the car's speed is \(20 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.3 \, \text{m s}^{-2}\).

1. Find the values of \(k\) and \(P\). [7]

The power is increased to 15 kW.

1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]

A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$

1. 1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
   2. Explain the significance of the sign of the answer to part (a)(i). [1]
2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]

A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).

1. Find
   [5]

The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.

1. Calculate the speed of the car at B. [6]

A car of mass 800 kg moves in a straight line along a horizontal road. There is a constant resistance to the motion of the car of magnitude 600 N. When the car is travelling at a speed of \(15 \text{ m s}^{-1}\) the power developed by the car is 27 kW. Determine the acceleration of the car when it is travelling at \(15 \text{ m s}^{-1}\). [4]

A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW. There is a constant resistance to motion of magnitude 600 N.

1. Find the greatest steady speed at which the car can travel. [2]

Later in the journey, while travelling at a speed of \(15 \text{ m s}^{-1}\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin^{-1}\left(\frac{1}{40}\right)\) to the horizontal. The power developed by the car remains constant at 18 kW. The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103 000 J. The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \text{ m s}^{-1}\).

1. Determine the distance the car travels from the bottom to the top of the hill. [5]

A car of mass \(900\) kg moves along a straight level road. The power developed by the car is constant and equal to \(60\) kW. The resistance to the motion of the car is constant and equal to \(1500\) N. At time \(t\) seconds the velocity of the car is denoted by \(v\) m s\(^{-1}\). Initially the car is at rest.

1. Show that \(\frac{3v\,dv}{5\,dt} = 40 - v\). [3]
2. Verify that \(t = 24\ln\left(\frac{40}{40-v}\right) - \frac{3}{5}v\). [5]

A tractor has a mass of 6000 kg. When developing a power of 5 kW, the tractor is travelling at a steady speed of 2.5 m s\(^{-1}\) across a horizontal field.

1. Calculate the magnitude of the resistance to the motion of the tractor. [2]

The tractor comes to horizontal ground where the resistance to motion is different. The power developed by the tractor during the next 10 s has an average value of 8 kW. During this time, the tractor accelerates uniformly from 2.5 m s\(^{-1}\) to 3 m s\(^{-1}\).

1. 1. Show that the work done against the resistance to motion during the 10 s is 71 750 J. [4]
   2. Assuming that the resistance to motion is constant, calculate its value. [3]

The tractor can usually travel up a straight track inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \frac{1}{20}\), while accelerating uniformly from 3 m s\(^{-1}\) to 3.25 m s\(^{-1}\) over a distance of 100 m against a resistance to motion of constant magnitude of 2000 N. The tractor develops a fault which limits its maximum power to 16kW.

1. Determine whether the tractor could now perform the same motion up the track. [You should assume that the mass of the tractor and the resistance to motion remain the same.] [7]

A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]

A vehicle of mass 3500 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal. When the vehicle is travelling at a velocity of \(v\text{ ms}^{-1}\), the resistance to motion can be modelled by a variable force of magnitude \(40v\) N.

1. Given that \(\sin\alpha = \frac{3}{49}\), calculate the power developed by the engine at the instant when the speed of the vehicle is \(25\text{ ms}^{-1}\) and its deceleration is \(0.2\text{ ms}^{-2}\). [5]
2. When the vehicle's engine is working at a constant rate of 40 kW, the maximum speed that can be maintained up the slope is \(20\text{ ms}^{-1}\). Find the value of \(\alpha\). Give your answer in degrees, correct to one significant figure. [5]

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]