A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car’s engine is constant. There is a constant resistance to motion of 650 N.
(a) Find the power of the car’s engine, given that the car’s acceleration is 0.5 m s-2 when its speed is 20 m s-1.
(b) Find the steady speed which the car can maintain with the engine working at this power.
A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N.
(a) Find, in kW, the rate at which the engine of the car is working when it is travelling at a constant speed of 20 m s-1.
(b) Find the acceleration of the car when its speed is 20 m s-1 and the engine is working at 15 kW.
A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N.
(a) Find the driving force when the acceleration of the minibus is 0.5 m s-2.
(b) Find the power required for the minibus to maintain a constant speed of 25 m s-1.
A car of mass 1250 kg is moving on a straight road.
(a) On a horizontal section of the road, the car has a constant speed of 32 m s-1 and there is a constant force of 750 N resisting the motion.
(b) On a section of the road inclined at sin-1 0.096 to the horizontal, the resistance to the motion of the car is (1000 + 8v) N when the speed of the car is v m s-1. The car travels up this section of the road at constant speed with the engine working at 60 kW.
Find this constant speed. [5]
A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at 1.5 \(\text{ms}^{-2}\). There are constant resistance forces of 250 N on the car and 100 N on the trailer.
(a) Find the tension in the tow-bar.
(b) Find the power of the engine of the car at the instant when the speed is 20 \(\text{ms}^{-1}\).
A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10 s is 750000 J.
(a) Find the power of the lorry’s engine.
(b) There is a constant resistance force acting on the lorry of magnitude 2400 N.
Find the acceleration of the lorry at an instant when its speed is 25 m s-1.
A car of mass 1300 kg is moving on a straight road.
(a) On a horizontal section of the road, the car has a constant speed of 30 m/s and there is a constant force of 650 N resisting the motion.
(b) On a section of the road inclined at \(\sin^{-1} 0.08\) to the horizontal, the resistance to the motion of the car is \((1000 + 20v)\) N when the speed of the car is \(v \text{ m/s}\). 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.
A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg. His power output is a constant 240 W. His acceleration when he is travelling at 6 m/s is 0.3 m/s2.
A crane is lifting a load of 1250 kg vertically at a constant speed \(V\) m s-1. Given that the power of the crane is a constant 20 kW, find the value of \(V\).
A car of mass 1400 kg is travelling up a hill inclined at an angle of 4° to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.
(i) Given that the engine of the car is working at 30 kW, find the speed of the car at an instant when its acceleration is 0.4 m s-2.
(ii) The greatest possible constant speed at which the car can travel up the hill is 40 m s-1. Find the maximum possible power of the engine.
A car has mass 1000 kg. When the car is travelling at a steady speed of \(v \text{ m s}^{-1}\), where \(v > 2\), the resistance to motion of the car is \((Av + B) \text{ N}\), where \(A\) and \(B\) are constants. The car can travel along a horizontal road at a steady speed of \(18 \text{ m s}^{-1}\) when its engine is working at \(36 \text{ kW}\). The car can travel up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\), at a steady speed of \(12 \text{ m s}^{-1}\) when its engine is working at \(21 \text{ kW}\). Find \(A\) and \(B\).
A lorry has mass 12,000 kg.
(i) The lorry moves at a constant speed of 5 m s-1 up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.08\). At this speed, the magnitude of the resistance to motion on the lorry is 1500 N. Show that the power of the lorry’s engine is 55.5 kW.
When the speed of the lorry is \(v\) m s-1 the magnitude of the resistance to motion is \(kv^2\) N, where \(k\) is a constant.
(ii) Show that \(k = 60\).
(iii) The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at 55.5 kW, find the lorry’s speed.
A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant speed of 20 m s-1. The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car’s engine is 6000 W. There are constant resistances to motion of R N on the car and 80 N on the trailer.
(i) Find the value of R.
The power of the car’s engine is increased to 12 500 W. The resistance forces do not change.
(ii) Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is 25 m s-1.
A van of mass 3200 kg travels along a horizontal road. The power of the van’s engine is constant and equal to 36 kW, and there is a constant resistance to motion acting on the van.
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.
The car comes to a hill inclined at 1° to the horizontal, still travelling at 15 m s-1.
A high-speed train of mass 490,000 kg is moving along a straight horizontal track at a constant speed of 85 m s-1. The engines are supplying 4080 kW of power.
(i) Show that the resistance force is 48,000 N.
(ii) The train comes to a hill inclined at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac{1}{200}\). Given that the resistance force is unchanged, find the power required for the train to keep moving at the same constant speed of 85 m s-1.
A car of mass 1400 kg travelling at a speed of \(v \text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v \text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56 \text{ m s}^{-1}\).
A lorry of mass 15,000 kg moves on a straight horizontal road in the direction from A to B. It passes A and B with speeds 20 m/s and 25 m/s respectively. The power of the lorry’s engine is constant and there is a constant resistance to motion of magnitude 6000 N. The acceleration of the lorry at B is 0.5 times the acceleration of the lorry at A.
(a) Show that the power of the lorry’s engine is 200 kW, and hence find the acceleration of the lorry when it is travelling at 20 m/s.
The lorry begins to ascend a straight hill inclined at 1° to the horizontal. It is given that the power of the lorry’s engine and the resistance force do not change.
(b) Find the steady speed up the hill that the lorry could maintain.
A train of mass 240,000 kg travels up a slope inclined at an angle of 4° to the horizontal. There is a constant resistance of magnitude 18,000 N acting on the train. At an instant when the speed of the train is 15 m/s, its deceleration is 0.2 m/s². Find the power of the engine of the train.
A car of mass 1200 kg has a greatest possible constant speed of 60 m s-1 along a straight level road. When the car is travelling at a speed of v m s-1 there is a resistive force of magnitude 35v N.