Up and down hill: two equations

5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .

1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A lorry of mass 7850 kg travels on a straight hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 1480 N .

1. Find the power of the lorry's engine when the lorry is going up the hill at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the power of the lorry's engine at an instant when the lorry is going down the hill at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. A car of mass 900 kg is moving on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 49 }\). When the car is moving up the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.

When the car is moving down the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is still modelled as a constant force of magnitude \(R\) newtons. Find

1. the value of \(R\),
2. the value of \(v\).

1. A truck of weight 9000 N is travelling up a hill on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\)

When the truck travels up the hill with the engine working at \(3 P\) watts, the truck is moving at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Later on, the truck travels down the hill along the same road, with the engine working at \(P\) watts. At the instant when the speed of the truck is \(12 \mathrm {~ms} ^ { - 1 }\), the acceleration of the truck is \(\frac { g } { 20 }\) The resistance to motion of the truck from non-gravitational forces is a constant force of magnitude \(R\) newtons in all circumstances. Find (i) the value of \(P\),
(ii) the value of \(R\).
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2 A car of mass 1400 kg , travels along a straight horizontal road AB , after which it descends a hill BC inclined at a constant angle of \(7 ^ { \circ }\) to the horizontal (see diagram). \(\mathrm { A } , \mathrm { B }\) and C all lie in the same vertical plane. Throughout the entire journey, the total resistance to the car's motion is constant. \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_232_1227_392_251} Between A and B, the car moves at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the power developed by the car is a constant \(P \mathrm {~W}\). When the car reaches B , the engine is switched off and the car travels down a line of greatest slope from \(B\) to \(C\) with an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is unchanged.

1. Determine the value of \(P\). When the car reaches C it turns round and travels back up the hill towards B at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power developed by the car between C and B is a constant 16 kW . The resistance to motion is unchanged.
2. Determine the value of \(v\).

A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]