Energy method - driving force up incline, find KE/PE changes as sub-parts

6 A block of mass 50 kg is pulled up a straight hill and passes through points \(A\) and \(B\) with speeds \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 200 m and \(B\) is 15 m higher than \(A\). For the motion of the block from \(A\) to \(B\), find

1. the loss in kinetic energy of the block,
2. the gain in potential energy of the block. The resistance to motion of the block has magnitude 7.5 N.
3. Find the work done by the pulling force acting on the block. The pulling force acting on the block has constant magnitude 45 N and acts at an angle \(\alpha ^ { \circ }\) upwards from the hill.
4. Find the value of \(\alpha\).

4 A lorry of mass 14000 kg moves along a road starting from rest at a point \(O\). It reaches a point \(A\), and then continues to a point \(B\) which it reaches with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The part \(O A\) of the road is straight and horizontal and has length 400 m . The part \(A B\) of the road is straight and is inclined downwards at an angle of \(\theta ^ { \circ }\) to the horizontal and has length 300 m .

1. For the motion from \(O\) to \(B\), find the gain in kinetic energy of the lorry and express its loss in potential energy in terms of \(\theta\). The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry from \(O\) to \(B\) is 5000 kJ .
2. Find the value of \(\theta\).

4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.

4 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and is inclined at \(4.5 ^ { \circ }\) to the horizontal. The speed of the block is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the slope and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the slope.

1. Find the loss of kinetic energy and the gain in potential energy of the block.
2. Given that the work done against the resistance to motion is 50 J , find the work done by the pulling force acting on the block.
3. Given also that the pulling force is constant and acts at an angle of \(15 ^ { \circ }\) upwards from the slope, find its magnitude.

5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find

1. the increase in kinetic energy,
2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
3. Calculate the distance \(A B\).