A lorry of mass 7850 kg travels on a straight hill which is inclined at an angle of 3° to the horizontal. There is a constant resistance to motion of 1480 N.
(i) Find the power of the lorry’s engine when the lorry is going up the hill at a constant speed of 10 m s-1.
(ii) Find the power of the lorry’s engine at an instant when the lorry is going down the hill at a speed of 15 m s-1 with an acceleration of 0.8 m s-2.
A cyclist is riding up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.04\). 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.
(i) Find the power output of the cyclist.
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.
(ii) 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.
A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of 12 m s-1. The total resistance to motion is 1150 N.
The tractor comes to a hill inclined at 4° above the horizontal. The power output is increased to 25 kW and the resistance to motion is unchanged.
A car of mass 1200 kg is travelling along a horizontal road.
(i) It is given that there is a constant resistance to motion.
(a) The engine of the car is working at 16 kW while the car is travelling at a constant speed of 40 m s-1. Find the resistance to motion.
(b) 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 m s-1.
(ii) It is given instead that the resistance to motion of the car is (590 + 2v) N when the speed of the car is v m s-1. The car travels at a constant speed with the engine working at 16 kW. Find this speed.
A car of mass 1200 kg is moving on a straight road against a constant force of 850 N resisting the motion.
(i) On a part of the road that is horizontal, the car moves with a constant speed of 42 m s-1.
(a) Calculate, in kW, the power developed by the engine of the car. [2]
(b) Given that this power is suddenly increased by 6 kW, find the instantaneous acceleration of the car. [3]
(ii) On a part of the road that is inclined at θ° 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 θ. [4]
A car of mass 900 kg is moving on a straight horizontal road ABCD. There is a constant resistance of magnitude 800 N in the sections AB and BC, and a constant resistance of magnitude R N in the section CD. The power of the car’s engine is a constant 36 kW.
A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant resistance to motion of 20 N. At an instant when the cyclist’s speed is 5 m s-1, his acceleration is 0.15 m s-2.
(i) Show that the total mass of the cyclist and bicycle is 80 kg.
The cyclist comes to a hill inclined at 2° to the horizontal. When the cyclist starts climbing the hill, he increases his power to a constant 300 W. The resistance to motion remains 20 N.
(ii) Show that the steady speed up the hill which the cyclist can maintain when working at this power is 6.26 m s-1, correct to 3 significant figures.
(iii) Find the acceleration at an instant when the cyclist is travelling at 90% of the speed in part (ii).
A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m. There is a constant resistance to motion of 25 N.
A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of 2 m s-1. There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.
(a) State the tension in the coupling.
(b) Find the power produced by the locomotive’s engine.
The power produced by the locomotive’s engine is now changed to 1.2 W.
(c) Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate.
A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of 25 m s-1. The system of the van and the trailer is modelled as two particles connected by a light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on the trailer.
(i) Find the power of the van’s engine.
(ii) Write down the tension in the cable.
The van reaches the bottom of a hill inclined at 4° to the horizontal with speed 25 m s-1. The power of the van’s engine is increased to 25 000 W.
(iii) Assuming that the resistance forces remain the same, find the new tension in the cable at the instant when the speed of the van up the hill is 20 m s-1.
The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N.
A car of mass 1100 kg is moving on a road against a constant force of 1550 N resisting the motion.
(i) The car moves along a straight horizontal road at a constant speed of 40 m s-1.
(ii) The car now travels at constant speed up a straight road inclined at 8° to the horizontal, with the engine working at 80 kW. Assuming the resistance force remains the same, find this constant speed. [3]
A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude 300 N.
(i) Find, in kW, the rate at which the engine of the car is working when the car has a constant speed of 40 m s-1.
(ii) Find the acceleration of the car when its speed is 25 m s-1 and the engine is working at 90% of the power found in part (i).
A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg.
A cyclist and his bicycle have a total mass of 90 kg. The cyclist starts to move with speed 3 m s-1 from the top of a straight hill, of length 500 m, which is inclined at an angle of sin-1 0.05 to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed 5 m s-1. The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, R N, and the cyclist’s speed, v m s-1, both vary.
A lorry of mass 24,000 kg is travelling up a hill which is inclined at 3° to the horizontal. The power developed by the lorry’s engine is constant, and there is a constant resistance to motion of 3200 N.
A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
(i) Find the work done by the weightlifter.
(ii) Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter.
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 m s\(^{-1}\) and acceleration 4 m s\(^{-2}\). The car passes through another point with speed 22.5 m s\(^{-1}\) and acceleration 0.3 m s\(^{-2}\). Find the values of \(P\) and \(R\).
A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and the block moves at a constant speed of 2.75 m s-1. Find the work done by the tension in 40 s and find the power applied by the tension.
A crate of mass 200 kg is being pulled at constant speed along horizontal ground by a horizontal rope attached to a winch. The winch is working at a constant rate of 4.5 kW and there is a constant resistance to the motion of the crate of magnitude 600 N.
(a) Find the time that it takes for the crate to move a distance of 15 m.
The rope breaks after the crate has moved 15 m.
(b) Find the time taken, after the rope breaks, for the crate to come to rest.