Find acceleration given power

1 A car of mass 880 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to motion is 700 N . At an instant when the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(P\).

3 A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude 300 N .

1. Find, in kW , the rate at which the engine of the car is working when the car has a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at \(90 \%\) of the power found in part (i).

1 A car of mass 1000 kg travels along a horizontal straight road with its engine working at a constant rate of 20 kW . The resistance to motion of the car is 600 N . Find the acceleration of the car at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 A motorcycle of mass 100 kg is travelling on a horizontal straight road. Its engine is working at a rate of 8 kW . At an instant when the speed of the motorcycle is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find, at this instant,

1. the force produced by the engine,
2. the resistance to motion of the motorcycle.

1 A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of 22.5 kW . Find the resistance to the car's motion at an instant when the car's speed is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When the car travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

1. Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
2. Find the maximum acceleration of the car when it is travelling at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) [0pt] [4 marks]

4 A car travels along a straight horizontal road. When its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car experiences a resistance force of magnitude \(25 v\) newtons.

1. The car has a maximum constant speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road. Show that the power being used to propel the car at this speed is 44100 watts.
2. The car has mass 1500 kg . Find the acceleration of the car when it is travelling at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road under a power of 44100 watts.

7 A train, of mass 22 tonnes, moves along a straight horizontal track. A constant resistance force of 5000 N acts on the train. The power output of the engine of the train is 240 kW . Find the acceleration of the train when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).

2 A car of mass 1200 kg is travelling in a straight line along a horizontal road. At a time when the power of the driving force is 25 kW , the car has a speed of \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the resistance to the motion of the car.

1. A lorry of mass 16000 kg moves along a straight horizontal road.

The lorry moves at a constant speed of \(25 \mathrm {~ms} ^ { - 1 }\) In an initial model for the motion of the lorry, the resistance to the motion of the lorry is modelled as having constant magnitude 16000 N .

1. Show that the engine of the lorry is working at a rate of 400 kW . The model for the motion of the lorry along the same road is now refined so that when the speed of the lorry along the same road is \(V \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the lorry is modelled as having magnitude 640 V newtons. Assuming that the engine of the lorry is working at the same rate of 400 kW
2. use the refined model to find the speed of the lorry when it is accelerating at \(2.1 \mathrm {~ms} ^ { - 2 }\)

4 A car has mass 1000 kg and travels on a straight horizontal road. The maximum speed of the car on this road is \(48 \mathrm {~ms} ^ { - 1 }\) In a simple model, it is assumed that the car experiences a resistance force that is proportional to its speed. When the car travels at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the resistance force is 600 newtons. 4

1. Show that the maximum power of the car is 69120 W
   4
2. Find the maximum acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) 4
3. Find the maximum acceleration of the car when it is travelling at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 4
4. Comment on the validity of the model in the context of your answers to parts (b) and (c).

1 A lorry moves along a straight horizontal road. The engine of the lorry produces a constant power of 80 kW . The mass of the lorry is 10 tonnes and the resistance to motion is constant at 4000 N .

1. Express the driving force of the lorry in terms of its velocity and hence, using Newton's second law, write down a differential equation which connects the velocity of the lorry and the time for which it has been moving.
2. Hence find the time taken, in seconds, for the lorry to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A car of mass \(1200\) kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to \(16\) kW. There is a constant resistance to motion of magnitude \(500\) N.

1. Find the acceleration of the car at an instant when its speed is \(20\) m s\(^{-1}\). [3]
2. Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel. [2]

The car comes to the bottom of a straight hill of length \(316\) m, inclined at an angle to the horizontal of \(\sin^{-1}(\frac{4}{65})\). The power remains constant at \(16\) kW, but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is \(128400\) J. The time taken to ascend the hill is \(15\) s.

1. Given that the car is travelling at a speed of \(20\) m s\(^{-1}\) at the bottom of the hill, find its speed at the top of the hill. [6]

A car of mass \(1200 \text{ kg}\) travels along a horizontal straight road. The power provided by the car's engine is constant and equal to \(20 \text{ kW}\). The resistance to the car's motion is constant and equal to \(500 \text{ N}\). The car passes through the points \(A\) and \(B\) with speeds \(10 \text{ m s}^{-1}\) and \(25 \text{ m s}^{-1}\) respectively. The car takes \(30.5 \text{ s}\) to travel from \(A\) to \(B\).

1. Find the acceleration of the car at \(A\). [4]
2. By considering work and energy, find the distance \(AB\). [8]

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 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 \(M\) 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 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]

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 \(900\) 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]