Acceleration from power and speed

1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .

The total mass of the cyclist and her cycle is 80 kg .
The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .

1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
   The distance from \(A\) to \(B\) is 20 km .
   Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
   The cyclist starts from rest at \(A\).
   At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
3. Using this model, find the value of \(P\)

5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 4 m above the level of \(A\).

1. Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
2. Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance \(A B\). At \(B\) the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
3. Find the acceleration with which the cyclist starts to move horizontally.

2 A car of mass 850 kg is travelling along a road that is straight but not level.
On one section of the road the car travels at constant speed and gains a vertical height of 60 m in 20 seconds. Non-gravitational resistances to its motion (e.g. air resistance) are negligible.

1. Show that the average power produced by the car is about 25 kW . On a horizontal section of the road, the car develops a constant power of exactly 25 kW and there is a constant resistance of 800 N to its motion.
2. Calculate the maximum possible steady speed of the car.
3. Find the driving force and acceleration of the car when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When travelling along the horizontal section of road, the car accelerates from \(15 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6.90 seconds with the same constant power and constant resistance.
4. By considering work and energy, find how far the car travels while it is accelerating. When the car is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a constant slope inclined at \(\arcsin ( 0.05 )\) to the horizontal, the driving force is removed. Subsequently, the resistance to the motion of the car remains constant at 800 N .
5. What is the speed of the car when it has travelled a further 105 m up the slope?

A car of mass 860 kg travels along a straight horizontal road. The power provided by the car's engine is \(P\) W and the resistance to the car's motion is \(R\) N. The car passes through one point with speed \(4.5 \text{ m s}^{-1}\) and acceleration \(4 \text{ m s}^{-2}\). The car passes through another point with speed \(22.5 \text{ m s}^{-1}\) and acceleration \(0.3 \text{ m s}^{-2}\). Find the values of \(P\) and \(R\). [6]

A car of mass \(1200\) kg is moving on a straight road against a constant force of \(850\) N resisting the motion.

1. On a part of the road that is horizontal, the car moves with a constant speed of \(42\) m s\(^{-1}\).
   1. Calculate, in kW, the power developed by the engine of the car. [2]
   2. Given that this power is suddenly increased by \(6\) kW, find the instantaneous acceleration of the car. [3]
2. On a part of the road that is inclined at \(\theta°\) to the horizontal, the car moves up the hill at a constant speed of \(24\) m s\(^{-1}\), with the engine working at \(80\) kW. Find \(\theta\). [4]

A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant rate of \(P\) kW. The resistance to motion of the car is 550 N. Given that the acceleration of the car is \(0.2 \text{ m s}^{-2}\) at an instant when its speed is \(30 \text{ m s}^{-1}\), find the value of \(P\). [4]

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 car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(j\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(j\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]

1. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(α\) with the horizontal, where \(\tan α = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{4}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(f\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(f\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]
2. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A car of mass 1200 kg moves along a straight horizontal road with a constant speed of 24 m s\(^{-1}\). The resistance to motion of the car has magnitude 600 N.

1. Find, in kW, the rate at which the engine of the car is working. [2]

The car now moves up a hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{20}\). The resistance to motion of the car from non-gravitational forces remains of magnitude 600 N. The engine of the car now works at a rate of 30 kW.

1. Find the acceleration of the car when its speed is 20 m s\(^{-1}\). [4]

A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{15}\). The resistance to motion of the car from non-gravitational forces has constant magnitude \(R\) newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of 20 m s\(^{-1}\).

1. Show that \(R = 260\). (4)

The power developed by the car's engine is now increased to 18 kW. The magnitude of the resistance to motion from non-gravitational forces remains at 260 N. At the instant when the car is moving up the road at 20 m s\(^{-1}\) the car's acceleration is \(a\) m s\(^{-2}\).

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