4 A car has a mass of 850 kg and its engine can generate a maximum power of 35 kW . The total resistance to motion of the car is modelled as \(k v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. When the car is moving in a straight line on a straight horizontal road, the greatest constant speed that it can attain is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 56\).
2. Find the greatest possible acceleration of the car on the road at an instant when it is moving with a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A trailer of mass 240 kg is attached to the car by means of a light inextensible tow bar which is parallel to the surface of the road. The resistance to motion of the trailer is modelled as a constant force of magnitude 350 N . The car and trailer move on the horizontal road. At a certain instant the car's engine is working at a rate of 30 kW and the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the speed of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. The car and trailer now move in a straight line on a straight road inclined at \(8 ^ { \circ }\) to the horizontal.
4. Find the difference between their greatest possible constant speed travelling up the slope and their greatest possible constant speed travelling down the slope.