Average power over journey

8 As part of an industrial process a single pump causes the intake of a liquid chemical to the bottom end of a tube, draws it up the tube and then discharges it through a nozzle at the top end of the tube. The tube is straight and narrow, 35 m long and inclined at an angle of \(26 ^ { \circ }\) to the horizontal. The chemical arrives at the intake at the bottom end of the tube with a speed of \(6.2 \mathrm {~ms} ^ { - 1 }\). At the top end of the tube the chemical is discharged horizontally with a speed of \(14.3 \mathrm {~ms} ^ { - 1 }\) (see diagram). In total, the pump discharges 1500 kg of chemical through the nozzle each hour. \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-5_405_1175_685_242} In order to model the changes to the mechanical energy of the chemical during the entire process of intake, drawing and discharge, the following modelling assumptions are made.

• At any instant the total resistance to the motion of all the liquid in the tube is 40 N .
• All other resistances to motion are ignored.
• The liquid in the tube moves at a constant speed of \(6.2 \mathrm {~ms} ^ { - 1 }\).
  1. 1. Find the difference between the total amount of energy output by the pump each hour and the total amount of mechanical energy gained by the chemical each hour.
     2. Give one reason why the model underestimates the power of the engine.

3 A pump is being used to empty a flooded basement.
In one minute, 400 litres of water are pumped out of the basement.
The water is raised 8 metres and is ejected through a pipe at a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
The mass of 400 litres of water is 400 kg .

1. Calculate the gain in potential energy of the 400 litres of water.
2. Calculate the gain in kinetic energy of the 400 litres of water.
3. Hence calculate the power of the pump, giving your answer in watts.

2 A car of mass 1200 kg travels along a road for two minutes during which time it rises a vertical distance of 60 m and does \(1.8 \times 10 ^ { 6 } \mathrm {~J}\) of work against the resistance to its motion. The speeds of the car at the start and at the end of the two minutes are the same.

1. Calculate the average power developed over the two minutes. The car now travels along a straight level road at a steady speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while developing constant power of 13.5 kW .
2. Calculate the resistance to the motion of the car. How much work is done against the resistance when the car travels 200 m ? While travelling at \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car starts to go down a slope inclined at \(5 ^ { \circ }\) to the horizontal with the power removed and its brakes applied. The total resistance to its motion is now 1500 N .
3. Use an energy method to determine how far down the slope the car travels before its speed is halved. Suppose the car is travelling along a straight level road and developing power \(P \mathrm {~W}\) while travelling at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of \(R \mathrm {~N}\).
4. Show that \(P = ( R + 1200 a ) v\) and deduce that if \(P\) and \(R\) are constant then if \(a\) is not zero it cannot be constant.

2. A pump raises water from a well 12 metres below the ground and ejects the water through a pipe of diameter 10 cm at a speed of \(6 \mathrm {~ms} ^ { - 1 }\). Given that the mass of \(1 \mathrm {~m} ^ { 3 }\) of water is 1000 kg ,

1. find, in terms of \(\pi\), the mass of water discharged by the pipe every second,
2. find in kJ , correct to 3 significant figures, the total mechanical energy gained by the water per second.

1 A clock is driven by a 5 kg sphere falling once through a vertical distance of 120 cm over 2 days. Calculate, in watts, the average power developed by the falling sphere.

2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.

A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when

1. \(t = 3\), [3]

1. \(t = 6\). [5]

Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at 8 m s\(^{-1}\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m. When she reaches the point \(B\), her speed is 5 m s\(^{-1}\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,

1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]

At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),

1. find the power generated by the cyclist at \(B\). [4]