91Ó°ÊÓ

A car of mass \(1000 \mathrm{~kg}\) is travelling on a level road against a resistance to motion which varies as the square of its speed. If the maximum power of the engine is \(60 \mathrm{~kW}\) and the car has a maximum speed of \(150 \mathrm{~km} / \mathrm{h}\), find an expression for the resistance to motion at any speed. Find also the acceleration when the engine is working at three-quarters full power and the speed is \(30 \mathrm{~km} / \mathrm{h}\).

A car is climbing a hill against a resistance to motion which is proportional to its speed. Find the maximum power of the car. (a) The car has a maximum speed of \(20 \mathrm{~ms}^{-1}\) up the hill and a maximum speed of \(40 \mathrm{~ms}^{-1}\) on the level. (b) The inclination of the hill is arcsin \(\frac{1}{10}\) to the horizontal. (c) The mass of the car is \(1000 \mathrm{~kg}\).

A car of mass \(1000 \mathrm{~kg}\) is moving on a level road at a steady speed of \(100 \mathrm{~km} / \mathrm{h}\) with its engine working at \(60 \mathrm{~kW}\). Calculate in newtons the total resistance to motion, which may be assumed to be constant. The engine is now disconnected, the brakes are applied, and the car comes to rest in 100 metres. Assuming that the total resistance remains the same, show that the retarding force of the brakes is about 1700 newtons. If the engine is still disconnected, find the distance the car would run up a hill of inclination arcsin \(\frac{1}{10}\) before coming to rest, starting at \(100 \mathrm{~km} / \mathrm{h}\) when the same resistance and braking force are operating.

When a car of mass \(M\) kilograms is ascending a hill of inclination \(\alpha\) against a constant resistance of \(R\) newtons, its engine is working at \(P\) kilowatts. Prove that: $$ 1000 P=(R+9.8 M \sin \alpha+M f) v $$ where \(v \mathrm{~m} / \mathrm{sec}\) is the velocity of the car and \(f \mathrm{~m} / \mathrm{sec}^{2}\) its acceleration. If \(\sin \alpha=\frac{1}{8}, P=20, R=400, M=500\), find the maximum speed attained by the car. When the cat has attained this speed the power of the engine is suddenly increased to 25 kilowatts. Show that the immediate acceleration is about \(\frac{1}{2} \mathrm{~m} / \mathrm{sec}^{2}\)