Energy method - driving force on inclined plane (down hill)

6 A lorry of mass 15000 kg climbs a hill of length 500 m at a constant speed. The hill is inclined at \(2.5 ^ { \circ }\) to the horizontal. The resistance to the lorry's motion is constant and equal to 800 N .

1. Find the work done by the lorry's driving force. On its return journey the lorry reaches the top of the hill with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and continues down the hill with a constant driving force of 2000 N . The resistance to the lorry's motion is again constant and equal to 800 N .
2. Find the speed of the lorry when it reaches the bottom of the hill.

5 A lorry of mass 16000 kg moves on a straight hill inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The length of the hill is 500 m .

1. While the lorry moves from the bottom to the top of the hill at constant speed, the resisting force acting on the lorry is 800 N and the work done by the driving force is 2800 kJ . Find the value of \(\alpha\).
2. On the return journey the speed of the lorry is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. While the lorry travels down the hill, the work done by the driving force is 2400 kJ and the work done against the resistance to motion is 800 kJ . Find the speed of the lorry at the bottom of the hill.
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2 A lorry of mass 15000 kg moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top to the bottom of a straight hill of length 900 m . The top of the hill is 18 m above the level of the bottom of the hill. The total work done by the resistive forces acting on the lorry, including the braking force, is \(4.8 \times 10 ^ { 6 } \mathrm {~J}\). Find

1. the loss in gravitational potential energy of the lorry,
2. the work done by the driving force. On reaching the bottom of the hill the lorry continues along a straight horizontal road against a constant resistance of 1600 N . There is no braking force acting. The speed of the lorry increases from \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(X\), where \(X\) is 2500 m from the bottom of the hill.
3. By considering energy, find the work done by the driving force of the lorry while it travels from the bottom of the hill to \(X\).

3 A van of mass 2500 kg descends a hill of length 0.4 km inclined at \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it descends the hill. Find the work done by the van's engine as it descends the hill.