Constant speed up/down incline

5 A railway engine of mass 75000 kg is moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.01\). The engine is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine is working at 960 kW . There is a constant force resisting the motion of the engine.

1. Find the resistance force.
   The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the power of the engine is now reduced to 900 kW . The resistance to motion is no longer constant, but in the next 60 s the work done against the resistance force is 46500 kJ .
2. Find the speed of the engine at the end of the 60 s .

2 A box of mass 5 kg is pulled at a constant speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) for 15 s up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.

1. Find the change in gravitational potential energy of the box.
2. Find the work done by the pulling force.

6 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-3_218_1280_1146_431} \(A B\) and \(B C\) are straight roads inclined at \(5 ^ { \circ }\) to the horizontal and \(1 ^ { \circ }\) to the horizontal respectively. \(A\) and \(C\) are at the same horizontal level and \(B\) is 45 m above the level of \(A\) and \(C\) (see diagram, which is not to scale). A car of mass 1200 kg travels from \(A\) to \(C\) passing through \(B\).

1. For the motion from \(A\) to \(B\), the speed of the car is constant and the work done against the resistance to motion is 360 kJ . Find the work done by the car's engine from \(A\) to \(B\). The resistance to motion is constant throughout the whole journey.
2. For the motion from \(B\) to \(C\) the work done by the driving force is 1660 kJ . Given that the speed of the car at \(B\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that its speed at \(C\) is \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. The car's driving force immediately after leaving \(B\) is 1.5 times the driving force immediately before reaching \(C\). Find, correct to 2 significant figures, the ratio of the power developed by the car's engine immediately after leaving \(B\) to the power developed immediately before reaching \(C\).

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

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. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
2. 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. 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 The diagram shows an electric winch raising two crates A and B , with masses 40 kg and 25 kg , respectively. The cable connecting the winch to A , and the cable connecting A to B may both be modelled as light and inextensible. Furthermore, it can be assumed that there are no resistances to motion. \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-4_499_300_447_246} Throughout the entire motion, the power \(P \mathrm {~W}\) developed by the winch is constant.
Crates A and B are both being raised at a constant speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) when the cable connecting A and B breaks. After the cable between A and B breaks, crate A continues to be raised by the winch. Crate A now accelerates until it reaches a new constant speed of \(( v + 3 ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Determine

• the value of \(v\),
• the value of \(P\).

10 The engine of a lorry of mass 4000 kg works at a constant rate of 75 kW . Resistance to motion is modelled by a constant resistive force. On a horizontal road the lorry travels at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the work done by the engine in travelling for 1 minute on the horizontal road.
2. The lorry travels at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope of angle \(\sin ^ { - 1 } 0.05\) to the horizontal. Find the value of \(v\).

A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at 60° to the horizontal. The winch pulls the load a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N. Find the work done by the winch. [3]

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

A cyclist is riding up a straight hill inclined at an angle \(α\) to the horizontal, where \(\sin α = 0.04\). The total mass of the bicycle and rider is 80 kg. The cyclist is riding at a constant speed of 4 m s\(^{-1}\). There is a force resisting the motion. The work done by the cyclist against this resistance force over a distance of 25 m is 600 J.

1. Find the power output of the cyclist. [4]

The cyclist reaches the top of the hill, where the road becomes horizontal, with speed 4 m s\(^{-1}\). The cyclist continues to work at the same rate on the horizontal part of the road.

1. Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work done by the cyclist during this period against the resistance force is 1200 J. [4]

A car has mass 550 kg. When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of 30 m s\(^{-1}\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of 25 m s\(^{-1}\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.

1. 1. Find the value of \(R\).
   2. Find the value of \(P\). [8]
2. Find the acceleration of the car when it travels along the straight horizontal road at 20 m s\(^{-1}\) with the engine working at 50 kW. [4]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]

1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

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 800 kg is moving at a constant speed of 15 m s\(^{-1}\) down a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{4}\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 900 N.

1. Find, in kW, the rate of working of the engine of the car. [4]

When the car is travelling down the road at 15 m s\(^{-1}\), the engine is switched off. The car comes to rest in time \(T\) seconds after the engine is switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 900 N.

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

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 girl and her bicycle have a combined mass of 64 kg. She cycles up a straight stretch of road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{14}\). She cycles at a constant speed of 5 m s\(^{-1}\). When she is cycling at this speed, the resistance to motion from non-gravitational forces has magnitude 20 N.

1. Find the rate at which the cyclist is working. [4]

She now turns round and comes down the same road. Her initial speed is 5 m s\(^{-1}\), and the resistance to motion is modelled as remaining constant with magnitude 20 N. She free-wheels down the road for a distance of 80 m. Using this model,

1. find the speed of the cyclist when she has travelled a distance of 80 m. [5]

The cyclist again moves down the same road, but this time she pedals down the road. The resistance is now modelled as having magnitude proportional to the speed of the cyclist. Her initial speed is again 5 m s\(^{-1}\) when the resistance to motion has magnitude 20 N.

1. Find the magnitude of the resistance to motion when the speed of the cyclist is 8 m s\(^{-1}\). [1]

The cyclist works at a constant rate of 200 W.

1. Find the magnitude of her acceleration when her speed is 8 m s\(^{-1}\). [4]

A car of mass 1000 kg moves with constant speed \(V\) m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is working at a rate of 12 kW. The resistance to motion from non-gravitational forces has magnitude 500 N. Find the value of \(V\). [5]