Work-energy over time interval

3 A train of mass 200000 kg moves on a horizontal straight track. It passes through a point \(A\) with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and later it passes through a point \(B\). The power of the train's engine at \(B\) is 1.2 times the power of the train's engine at \(A\). The driving force of the train's engine at \(B\) is 0.96 times the driving force of the train's engine at \(A\).

1. Show that the speed of the train at \(B\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. For the motion from \(A\) to \(B\), find the work done by the train's engine given that the work done against the resistance to the train's motion is \(2.3 \times 10 ^ { 6 } \mathrm {~J}\).

7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.

1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
3. the distance \(A B\). {www.cie.org.uk} after the live examination series. }

1 A bus of mass 8 tonnes is driven up a hill on a straight road. On one part of the hill, the power of the driving force on the bus is constant at 20 kW for one minute.

1. Calculate how much work is done by the driving force in this time. During this minute the speed of the bus increases from \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\) and, in addition to the work done against gravity, 125000 J of work is done against the resistance to motion of the bus parallel to the slope.
2. Calculate the change in the kinetic energy of the bus.
3. Calculate the vertical displacement of the bus. On another stretch of the road, a driving force of power 26 kW is required to propel the bus up a slope of angle \(\theta\) to the horizontal at a constant speed of \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), against a resistance to motion of 225 N parallel to the slope.
4. Calculate the angle \(\theta\). The bus later travels up the same slope of angle \(\theta\) to the horizontal at the same constant speed of \(6.5 \mathrm {~ms} ^ { - 1 }\) but now against a resistance to motion of 155 N parallel to the slope.
5. Calculate the power of the driving force on the bus.

5 A car of mass 4000 kg travels up a line of greatest slope of a straight road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.1\).
The power developed by the car's engine is constant and the resistance to the motion of the car is constant and equal to 850 N . The car passes through a point A on the road with speed \(18 \mathrm {~ms} ^ { - 1 }\) and acceleration \(0.75 \mathrm {~ms} ^ { - 2 }\).

1. Calculate the power developed by the car. The car later passes through a point B on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car takes 17.8 s to travel from A to B .
2. Find the distance AB .

1 A car of mass 1500 kg travels along a horizontal straight road. There are no resistances to the car's motion. The power developed by the car as it increases its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over \(t\) seconds is a constant 5000 W .

1. Determine the value of \(t\).
2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A car of mass \(900\) kg is moving on a straight horizontal road \(ABCD\). There is a constant resistance of magnitude \(800\) N in the sections \(AB\) and \(BC\), and a constant resistance of magnitude \(R\) N in the section \(CD\). The power of the car's engine is a constant \(36\) kW.

1. The car moves from \(A\) to \(B\) at a constant speed in \(120\) s. Find the speed of the car and the distance \(AB\). [3]
2. The distance \(BC\) is \(450\) m. Find the speed of the car at \(C\). [3]
3. The car comes to rest at \(D\). The distance \(AD\) is \(6637.5\) m. Find the deceleration of the car and the value of \(R\). [4]

The car's engine is switched off at \(B\).