Variable resistance: find constant speed

6 A car of mass 1300 kg is moving on a straight road.

1. On a horizontal section of the road, the car has a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and there is a constant force of 650 N resisting the motion.
   1. Calculate, in kW , the power developed by the engine of the car.
   2. Given that this power is suddenly increased by 9 kW , find the instantaneous acceleration of the car.
2. On a section of the road inclined at \(\sin ^ { - 1 } 0.08\) to the horizontal, the resistance to the motion of the car is \(( 1000 + 20 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels downwards along this section of the road at constant speed with the engine working at 11.5 kW . Find this constant speed.

6 A car of mass 1200 kg is travelling along a horizontal road.

1. It is given that there is a constant resistance to motion.
   1. The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
   2. The power is now increased to 22.5 kW . Find the acceleration of the car at the instant it is travelling at a speed of \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   3. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.

2. A truck of mass 1800 kg is moving along a straight horizontal road. The engine of the truck is working at a constant rate of 10 kW . The non-gravitational resistance to motion is modelled as a constant force of magnitude \(R\) newtons. At the instant when the truck is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\).
   (4) The truck now moves up a straight road at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The non-gravitational resistance to motion is now modelled as a constant force of magnitude 30 V newtons. The engine of the truck is now working at a constant rate of 12 kW .
2. Find the value of \(V\).
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1. A motorcyclist and his motorcycle have a combined mass of 480 kg .

The motorcyclist drives down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 12 }\), with the engine of the motorcycle working at 3.5 kW . The motorcycle is moving at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the motorcycle is modelled as a constant force with magnitude 20 V newtons. Find the value of \(V\).
(5)

6 A motorbike and its rider, together denoted by \(M\), have a combined mass of 360 kg . The resistive force experienced by \(M\) when it is in motion is modelled as being proportional to the speed it is moving at. All motion of \(M\) is on a straight horizontal road. It is found that with the engine of the motorbike working at a rate of 12 kW , the maximum constant speed that \(M\) can move at is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the speed of \(M\) such that with the engine working at a rate of 12 kW the acceleration of \(M\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

2 A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW . In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R N\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 430\).
2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be 60 v where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.

1. A van of mass 750 kg is moving along a straight horizontal road. At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(\lambda \nu \mathrm { N }\), where \(\lambda\) is a constant.

The engine of the van is working at a constant rate of 18 kW .
At the instant when \(v = 15\), the acceleration of the van is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Show that \(\lambda = 50\) The van now moves up a straight road inclined at an angle to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\) At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v . When the engine of the van is working at a constant rate of 12 kW , the van is moving at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

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1. A racing car of mass 750 kg is moving along a straight horizontal road at a constant speed of \(U \mathbf { k m ~ h } ^ { - \mathbf { 1 } }\). The engine of the racing car is working at a constant rate of 60 kW .

The resistance to the motion of the racing car is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. Using the model,

1. find the value of \(U\) Later on, the racing car is accelerating up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 5 } { 49 }\). The engine of the racing car is working at a constant rate of 60 kW . The total resistance to the motion of the racing car from non-gravitational forces is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. At the instant when the acceleration of the racing car is \(2 \mathrm {~ms} ^ { - 2 }\), the speed of the racing car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model,
2. find the value of \(V\)

1. A lorry has mass 5000 kg .

In all circumstances, when the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion of the lorry from non-gravitational forces is modelled as having magnitude \(490 v\) newtons. The lorry moves along a straight horizontal road at \(12 \mathrm {~ms} ^ { - 1 }\), with its engine working at a constant rate of 84 kW . Using the model,

1. find the acceleration of the lorry. Another straight road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 14 }\) With its engine again working at a constant rate of 84 kW , the lorry can maintain a constant speed of \(V \mathrm {~ms} ^ { - 1 }\) up the road. Using the model,
2. find the value of \(V\).

1. A truck of mass 1200 kg is moving along a straight horizontal road.

At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck is modelled as a force of magnitude \(( 900 + 9 v ) \mathrm { N }\). The engine of the truck is working at a constant rate of 25 kW .

1. Find the deceleration of the truck at the instant when \(v = 25\) Later on, the truck is moving up a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck from non-gravitational forces is modelled as a force of magnitude ( \(900 + 9 v\) ) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(V\).

1. A van of mass 900 kg is moving along a straight horizontal road.

At the instant when the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). When the engine of the van is working at a constant rate of 18 kW , the van is moving along the road at a constant speed \(V \mathrm {~ms} ^ { - 1 }\)

1. Find the value of \(V\). Later on, the van is moving up a straight road that is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 1 } { 21 }\) At the instant when the speed of the van is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). The engine of the van is again working at a constant rate of 18 kW .
2. Find the acceleration of the van at the instant when \(v = 15\)

4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.

1. Show that the maximum power of the car is 52920 W .
2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
   1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
   2. Hence find \(V\).

4 A van, of mass 1500 kg , has a maximum speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. When the van travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(40 v\) newtons.

1. Show that the maximum power of the van is 100000 watts.
2. The van is travelling along a straight horizontal road. Find the maximum possible acceleration of the van when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The van starts to climb a hill which is inclined at \(6 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the van as it travels in a straight line up the hill.
   (6 marks)

5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.

2 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road using maximum power, its acceleration is \(3.3 \mathrm {~ms} ^ { - 2 }\). In an initial model of the motion of the car it is assumed that the resistance to motion is constant.

1. Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
2. Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be \(21.6 \mathrm {~ms} ^ { - 1 }\).
3. Explain what this value means about the models used in parts (a) and (b). \includegraphics[max width=\textwidth, alt={}, center]{aa25b8a6-9a5a-4de2-9534-18db8a175c34-03_583_378_169_255} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm { rads } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).
   1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
   2. Show that
      1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
      2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
      3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
   3. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.

A car of mass \(900\text{kg}\) is moving up a hill inclined at \(\sin^{-1} 0.12\) to the horizontal. The initial speed of the car is \(11\text{ms}^{-1}\). After \(12\text{s}\), the car has travelled \(150\text{m}\) up the hill and has speed \(16\text{ms}^{-1}\). The engine of the car is working at a constant rate of \(24\text{kW}\).

1. Find the work done against the resistive forces during the \(12\text{s}\). [5]
2. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \((1520 + 4v)\text{N}\) when the speed of the car is \(v\text{ms}^{-1}\). The car travels at a constant speed with the engine working at a constant rate of \(32\text{kW}\). Find this speed. [3]

A car has mass \(1200\) kg. When the car is travelling at a speed of \(v \text{ ms}^{-1}\), there is a resistive force of magnitude \(kv\) N. The maximum power of the car's engine is \(92.16\) kW.

1. The car travels along a straight level road.
   1. The car has a greatest possible constant speed of \(48 \text{ ms}^{-1}\). Show that \(k = 40\). [1]
   2. At an instant when its speed is \(45 \text{ ms}^{-1}\), find the greatest possible acceleration of the car. [3]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\sin^{-1} 0.15\) to the horizontal. Find the greatest possible speed of the car going up the hill. [4]

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]

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 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]