6.02m Variable force power: using scalar product

A car of mass \(1200\) kg is driving along a straight horizontal road at a constant speed of \(15\) m s\(^{-1}\). There is a constant resistance to motion of \(350\) N.

1. Find the power of the car's engine. [1]

The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).

1. The car starts to descend the hill with reduced power and with an acceleration of \(0.12\) m s\(^{-2}\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. [3]
2. When the car is travelling at \(20\) m s\(^{-1}\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains \(350\) N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18\) m s\(^{-1}\). [4]

A particle \(P\) of mass \(0.5\) kg moves in a straight line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The only forces acting on \(P\) are a force of magnitude \(\frac{150}{(x+1)^2}\) N in the direction of increasing displacement and a resistive force of magnitude \(\frac{450}{(x+1)^3}\) N. When \(t = 0\), \(x = 0\) and \(v = 20\). Find \(v\) in terms of \(x\), giving your answer in the form \(v = \frac{Ax + B}{(x + 1)}\), where \(A\) and \(B\) are constants to be determined. [6]

A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.

1. Find the speed and direction of \(C\) after the collision. [4]
2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
3. State a modelling assumption which you have made about the trucks in your solution [1]

Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.

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

A car of mass 1000 kg is moving along a straight horizontal road. The resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The engine of the car is working at a rate of 12 kW. When the car is moving with speed 15 m s\(^{-1}\), the acceleration of the car is 0.2 m s\(^{-2}\).

1. Show that \(R = 600\). [4]

The car now moves with constant speed \(U\) m s\(^{-1}\) downhill on a straight road inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is now working at a rate of 7 kW. The resistance to motion from non-gravitational forces remains of magnitude \(R\) newtons.

1. Calculate the value of \(U\). [5]

A car of mass 1000 kg is moving at a constant speed of 16 m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N.

1. Show that \(\sin \theta = \frac{1}{14}\). [5]

When the car is travelling up the road at 16 m s\(^{-1}\), the engine is switched off. The car comes to rest, without braking, having moved a distance \(y\) metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N.

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

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]

A lorry of mass 4200 kg can develop a maximum power of 84 kW. On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at 20 ms\(^{-1}\) the resisting force has magnitude 2400 N. Find the maximum speed of the lorry when it is

1. travelling on a horizontal road, [4 marks]
2. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{7}\). [6 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]

Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}

1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
   1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
   2. Calculate the work done against friction per tile. [2]
   3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]

A rocket, with initial mass 1500 kg, including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of 15 kg s\(^{-1}\) and the burnt fuel is ejected vertically downwards with a speed of 1000 m s\(^{-1}\) relative to the rocket. At time \(t\) seconds after launch \((t \leqslant 40)\) the rocket has mass \(m\) kg and velocity \(v\) m s\(^{-1}\).

1. Show that $$\frac{dv}{dt} + \frac{1000}{m} \frac{dm}{dt} = -9.8$$ [5]

1. Find \(v\) at time \(t\), \(0 \leqslant t \leqslant 40\) [5]

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 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 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 car of mass \(1250\) kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v\) 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 \(15\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.54\) m s\(^{-2}\). At a point \(B\) on the road the car's speed is \(20\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.3\) 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 van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{16}\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of 20 m s\(^{-1}\), the van's engine is working at a constant rate of 25 kW.

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

The power developed by the van's engine is now increased to 30 kW. The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at 20 m s\(^{-1}\), the acceleration of the van is \(a\) m s\(^{-2}\).

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

A car of mass 850 kg is being driven uphill along a straight road inclined at \(7°\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N. At a certain instant the car's speed is \(12 \text{ms}^{-1}\) and its acceleration is \(0.4 \text{ms}^{-2}\).

1. Calculate the power of the car's engine at this instant. [3]
2. Find the constant speed at which the car could travel up the hill with the engine generating this power. [2]

An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was

1. exerting a constant tractive force, [2]
2. working at constant power. [9]