7. A cyclist is pedalling along a horizontal cycle track at a constant speed of \(5 \mathrm {~ms} ^ { - 1 }\). The air resistance opposing her motion has magnitude 42 N . The combined mass of the cyclist and her machine is 84 kg .
- Find the rate, in W , at which the cyclist is working.
The cyclist now starts to ascend a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\), at a constant speed.
She continues to work at the same rate as before, against the same air resistance. - Find the constant speed at which she ascends the hill.
In fact the air resistance is not constant, and a revised model takes account of this by assuming that the air resistance is proportional to the cyclist's speed.
- Use this model to find an improved estimate of the speed at which she ascends the hill, if her rate of working still remains constant.