Find steady/maximum speed given power

2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .

1. The car is moving along a straight level road at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
2. The car travels at a constant speed down a hill inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 20 }\), with the engine working at 31.5 kW . Find the speed of the car.

3 A car travels along a horizontal straight road with increasing speed until it reaches its maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and equal to \(R \mathrm {~N}\), and the power provided by the car's engine is 18 kW .

1. Find the value of \(R\).
2. Given that the car has mass 1200 kg , find its acceleration at the instant when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .

1. The car is moving along a straight level road at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
2. The car travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.1\), with the engine working at 76.5 kW . Find this speed.

2 A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to motion is 1150 N .

1. Find the power output of the tractor's engine.
   The tractor comes to a hill inclined at \(4 ^ { \circ }\) above the horizontal. The power output is increased to 25 kW and the resistance to motion is unchanged.
2. Find the deceleration of the tractor at the instant it begins to climb the hill.
3. Find the constant speed that the tractor could maintain on the hill when working at this power.

1. A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg . The resistance to motion is modelled as a constant force of magnitude 32 N . The rate at which the cyclist works is 384 W . The cyclist accelerates until he reaches a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find

1. the value of \(v\),
2. the acceleration of the cyclist at the instant when the speed is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3. A truck of mass of 300 kg moves along a straight horizontal road with a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion of the truck has magnitude 120 N .

1. Find the rate at which the engine of the truck is working. On another occasion the truck moves at a constant speed up a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the truck from non-gravitational forces remains of magnitude 120 N . The rate at which the engine works is the same as in part (a).
2. Find the speed of the truck.

3 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~ms} ^ { - 1 }\) 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).

1 A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW .

1. The car can travel at constant speed \(v \mathrm {~ms} ^ { - 1 }\) along the road. Find \(v\).
2. Find the acceleration of the car at an instant when its speed is \(30 \mathrm {~ms} ^ { - 1 }\).

2 A car of mass 1600 kg moves along a straight horizontal road. The resistance to the motion of the car has constant magnitude 800 N and the car's engine is working at a constant rate of 20 kW .

1. Find the acceleration of the car at an instant when the car's speed is \(20 \mathrm {~ms} ^ { - 1 }\). The car now moves up a hill inclined at \(4 ^ { \circ }\) to the horizontal. The car's engine continues to work at 20 kW and the magnitude of the resistance to motion remains at 800 N .
2. Find the greatest steady speed at which the car can move up the hill.

3 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors and \(c\) is a positive real number.
The resultant of two forces \(c \mathbf { i N }\) and \(- \mathbf { i } + 2 \sqrt { c } \mathbf { j N }\) is denoted by \(R \mathrm {~N}\).

1. Show that the magnitude of \(R\) is \(c + 1\). A car of mass 900 kg travels along a straight horizontal road with constant resistance to motion of magnitude \(( c + 1 ) \mathrm { N }\). The car passes through point A on the road with speed \(6 \mathrm {~ms} ^ { - 1 }\), and 8 seconds later passes through a point B on the same road. The power developed by the car while travelling from A to B is zero. Furthermore, while travelling between A and B, the car's direction of motion is unchanged.
2. Determine the range of possible values of \(c\). The car later passes through a point C on the road. While travelling between B and C the power developed by the car is modelled as constant and equal to 18 kW . The car passes through C with speed \(5 \mathrm {~ms} ^ { - 1 }\) and acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\).
3. Determine the value of \(c\).
4. Suggest how one of the modelling assumptions made in this question could be improved.

2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

Two racing cars \(A\) and \(B\) are at rest alongside each other at a point \(O\) on a straight horizontal test track. The mass of \(A\) is 1200 kg. The engine of \(A\) produces a constant driving force of 4500 N. When \(A\) arrives at a point \(P\) its speed is 25 m s\(^{-1}\). The distance \(OP\) is \(d\) m. The work done against the resistance force experienced by \(A\) between \(O\) and \(P\) is 75 000 J.

1. Show that \(d = 100\). [3]

Car \(B\) starts off at the same instant as car \(A\). The two cars arrive at \(P\) simultaneously and with the same speed. The engine of \(B\) produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N.

1. Find the mass of \(B\). [3]
2. Find the steady speed which \(B\) can maintain when its engine is working at the same rate as it is at \(P\). [3]

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 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW.

1. The car can travel at constant speed \(v \text{ ms}^{-1}\) along the road. Find \(v\). [2]
2. Find the acceleration of the car at an instant when its speed is \(30 \text{ ms}^{-1}\). [3]