Find acceleration on incline given power

2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N . The power of the car's engine is 22500 W .

1. Show that the speed of the car is \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The car, moving with speed \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), comes to a section of the hill which is inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.

2 The total mass of a cyclist and his cycle is 80 kg . The resistance to motion is zero.

1. The cyclist moves along a horizontal straight road working at a constant rate of \(P \mathrm {~W}\). Find the value of \(P\) given that the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when his acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The cyclist moves up a straight hill inclined at an angle \(\alpha\), where \(\sin \alpha = 0.035\). Find the acceleration of the cyclist at an instant when he is working at a rate of 450 W and has speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A train of mass 240000 kg travels up a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance of magnitude 18000 N acting on the train. At an instant when the speed of the train is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its deceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power of the engine of the train.

3 A car of mass 1400 kg is travelling up a hill inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.

1. 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 \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The greatest possible constant speed at which the car can travel up the hill is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the maximum possible power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-06_643_419_255_863} Two particles \(A\) and \(B\), of masses 1.3 kg and 0.7 kg respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is 1.75 m above the floor and particle \(B\) is 1 m above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.

3 A cyclist travels along a straight road working at a constant rate of 420 W . The total mass of the cyclist and her cycle is 75 kg . Ignoring any resistance to motion, find the acceleration of the cyclist at an instant when she is travelling at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. given that the road is horizontal,
2. given instead that the road is inclined at \(1.5 ^ { \circ }\) to the horizontal and the cyclist is travelling up the slope.

3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .

1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .

1. A truck of mass 1500 kg is moving on a straight horizontal road.

The engine of the truck is working at a constant rate of 30 kW .
The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the truck is moving at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the value of \(R\). Later on, the truck is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 8 }\) The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 500 N .
   The engine of the truck is again working at a constant rate of 30 kW . At the instant when the speed of the truck is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(V\)

1. A car of mass 1500 kg is moving up a straight road, which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to the motion of the car from non-gravitational forces is constant and is modelled as a single constant force of magnitude 650 N . The car's engine is working at a rate of 30 kW .

Find the acceleration of the car at the instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. A van of mass 900 kg is moving down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to motion of the van has constant magnitude 570 N . The engine of the van is working at a constant rate of 12.5 kW .

At the instant when the van is moving down the road at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).

2 The power developed by the engine of a car as it travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road is 20 kW .

1. Calculate the resistance to the motion of the car. The car, of mass 1500 kg , now travels down a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is unchanged.
2. Find the power produced by the engine of the car when the car has speed \(32 \mathrm {~ms} ^ { - 1 }\) and is accelerating at \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. A car of mass 1200 kg moves up a straight road that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\)

The total resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. At the instant when the engine of the car is working at a rate of 32 kW and the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is \(0.5 \mathrm {~ms} ^ { - 2 }\) Find the value of \(R\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of \(P \mathrm {~kW}\).
Using the model,

1. find the value of \(P\). The car now travels in a straight line up a road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 49 }\)
   • In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude \(20 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car
   At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at \(10 \mathrm {~ms} ^ { - 1 }\), the acceleration of the car is \(a \mathrm {~ms} ^ { - 2 }\) Using the refined model,
2. find the value of \(a\). Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the refined model,
3. find the value of \(U\).

1. A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .

1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
   At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(w\).

A car of mass 1150 kg travels up a straight hill inclined at 1.2° to the horizontal. The resistance to motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed 16 m s\(^{-1}\) and the engine is working at a power of 35 kW. [4]

A car of mass \(1150 \text{ kg}\) travels up a straight hill inclined at \(1.2°\) to the horizontal. The resistance to motion of the car is \(975 \text{ N}\). Find the acceleration of the car at an instant when it is moving with speed \(16 \text{ m s}^{-1}\) and the engine is working at a power of \(35 \text{ kW}\). [4]

The maximum power produced by the engine of a small aeroplane of mass 2 tonnes is 128 kW. Air resistance opposes the motion directly and the lift force is perpendicular to the direction of motion. The magnitude of the air resistance is proportional to the square of the speed and the maximum steady speed in level flight is \(80 \text{ ms}^{-1}\).

1. Calculate the magnitude of the air resistance when the speed is \(60 \text{ ms}^{-1}\). [5]

The aeroplane is climbing at a constant angle of \(2°\) to the horizontal.

1. Find the maximum acceleration at an instant when the speed of the aeroplane is \(60 \text{ ms}^{-1}\). [4]

A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).

1. Find
   [5]

The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.

1. Calculate the speed of the car at B. [6]

A vehicle of mass 3500 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal. When the vehicle is travelling at a velocity of \(v\text{ ms}^{-1}\), the resistance to motion can be modelled by a variable force of magnitude \(40v\) N.

1. Given that \(\sin\alpha = \frac{3}{49}\), calculate the power developed by the engine at the instant when the speed of the vehicle is \(25\text{ ms}^{-1}\) and its deceleration is \(0.2\text{ ms}^{-2}\). [5]
2. When the vehicle's engine is working at a constant rate of 40 kW, the maximum speed that can be maintained up the slope is \(20\text{ ms}^{-1}\). Find the value of \(\alpha\). Give your answer in degrees, correct to one significant figure. [5]