Power-velocity relationship

3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, \(P\) watts, modelled by $$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$ where \(m \mathrm {~kg}\) is the mass of the aeroplane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity at time \(t\) seconds. The displacement of the aeroplane from its starting point is \(x \mathrm {~m}\). To take off successfully the aeroplane must reach a speed of \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it has travelled 900 m .

1. Formulate and solve a differential equation for \(v\) in terms of \(x\). Hence show that the aeroplane takes off successfully.
2. Formulate a differential equation for \(v\) in terms of \(t\). Solve the differential equation to show that \(v = 100 \tan ( 0.04 t )\). What feature of this result casts doubt on the validity of the model?
3. In fact the model is only valid for \(0 \leqslant t \leqslant 11\), after which the power remains constant at the value attained at \(t = 11\). Will the aeroplane take off successfully?

3 A car of mass 800 kg moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds. While \(v \leqslant 20\), the power developed by the engine is \(8 v ^ { 4 } \mathrm {~W}\). The total resistance force on the car is of magnitude \(8 v ^ { 2 } \mathrm {~N}\). Initially \(v = 2\) and the car is at a point O . At time \(t\) s the displacement from O is \(x \mathrm {~m}\).

1. Find \(v\) in terms of \(x\) and show that when \(v = 20 , x = 100 \ln 1.9\).
2. Find the relationship between \(t\) and \(x\), and show that when \(v = 20 , t \approx 19.2\). The driving force is removed at the instant when \(v\) reaches 20 .
3. For the subsequent motion, find \(v\) in terms of \(t\). Calculate \(t\) when \(v = 2\).

11 A car of mass 800 kg has a constant power output of 32 kW while travelling on a horizontal road. At time \(t \mathrm {~s}\) the car's speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistive force has magnitude \(20 v \mathrm {~N}\).

1. Show that \(v\) satisfies the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1600 - v ^ { 2 } } { 40 v }\).
2. Given that \(v = 0\) when \(t = 0\), solve this differential equation to find \(v\) in terms of \(t\). State what the solution predicts as \(t\) becomes large.

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).