Cyclist or runner: find resistance or speed

2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W . At an instant when her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is 20 N .

1. Find the total mass of the cyclist and her bicycle.
   The cyclist comes to a straight hill inclined at an angle \(\theta\) above the horizontal. She ascends the hill at constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She continues to work at the same rate as before and the resistance force is unchanged.
2. Find the value of \(\theta\).

4 The total mass of a cyclist and her bicycle is 70 kg . The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). At an instant when the cyclist's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude \(F \mathrm {~N}\).

1. Find the value of \(F\).
2. Find the steady speed that the cyclist could maintain up the hill when working at this power. [2]

3 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and his bicycle is 90 kg . The power exerted by the cyclist is 250 W . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the value of the constant resistance to motion acting on the cyclist.
   The cyclist comes to the bottom of a hill inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance to motion are unchanged, find the steady speed which the cyclist could maintain when riding up the hill.

1 A racing cyclist, whose mass with his cycle is 75 kg , works at a rate of 720 W while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to \(R \mathrm {~N}\).

1. Given that the cyclist is accelerating at \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when his speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(R\).
2. Given that the cyclist's acceleration is positive, show that his speed is less than \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 The resistance to motion acting on a runner of mass 70 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the runner's speed and \(k\) is a constant. The greatest power the runner can exert is 100 W . The runner's greatest steady speed on horizontal ground is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 6.25\).
2. Find the greatest steady speed of the runner while running uphill on a straight path inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).

5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the resistance to the cyclist's motion is 16 N .
2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).

3. A cyclist and his bicycle, with a combined mass of 75 kg , move along a straight horizontal road. The cyclist is working at a constant rate of 180 W . There is a constant resistance to the motion of the cyclist and his bicycle of magnitude \(R\) newtons. At the instant when the speed of the cyclist is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). Later, the cyclist moves up a straight road with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 21 }\). The cyclist is working at a rate of 180 W and the resistance to the motion of the cyclist and his bicycle from non-gravitational forces is again the same constant force of magnitude \(R\) newtons.
2. Find the value of \(v\).

1. A cyclist and his bicycle have a combined mass of 90 kg . He rides on a straight road up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\). He works at a constant rate of 444 W and cycles up the hill at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the magnitude of the resistance to motion from non-gravitational forces as he cycles up the hill.

4 A rower is rowing a boat in a straight line across a lake. The combined mass of the rower, boat and oars is 240 kg . The maximum power that the rower can generate is 450 W . In a model of the motion of the boat it is assumed that the total resistance to the motion of the boat is 150 N at any instant when the boat is in motion.

1. Find the maximum possible acceleration of the boat, according to the model, at an instant when its speed is \(0.5 \mathrm {~ms} ^ { - 1 }\). At one stage in its motion the boat is travelling at a constant speed and the rower is generating power at an average rate of 210 W , which is assumed to be constant. The boat passes a pole and then, after travelling 350 m , a second pole.
2. Determine how long it takes, according to the model, for the boat to travel between the two poles.
3. State a reason why the assumption that the rower's generated power is constant may be unrealistic.

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 .

1. 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.
2. 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.
3. 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.

1. \hspace{0pt} [In this question use \(\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ]

A jogger of mass 60 kg runs along a straight horizontal road at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to the motion of the jogger is modelled as a constant force of magnitude 30 N .

1. Find the rate at which the jogger is working. The jogger now comes to a hill which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). Because of the hill, the jogger reduces her speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and maintains this constant speed as she runs up the hill. The total resistance to the motion of the jogger from non-gravitational forces continues to be modelled as a constant force of magnitude 30 N .
2. Find the rate at which she has to work in order to run up the hill at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A cyclist and her bicycle have a total mass of \(70\) kg. She cycles along a straight horizontal road with constant speed \(3.5 \text{ ms}^{-1}\). She is working at a constant rate of \(490\) W.

1. Find the magnitude of the resistance to motion. [4]

The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), at a constant speed \(U \text{ ms}^{-1}\). The magnitude of the non-gravitational resistance to motion is modelled as \(40U\) newtons. She is now working at a constant rate of \(24\) W.

1. Find the value of \(U\). [7]

A cyclist and her cycle have a combined mass of \(75\) kg. The cyclist is cycling up a straight road inclined at \(5°\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude \(20\) N. At the instant when the cyclist has a speed of \(12\) m s\(^{-1}\), she is decelerating at \(0.2\) m s\(^{-2}\).

1. Find the rate at which the cyclist is working at this instant. [5]

When the cyclist passes the point \(A\) her speed is \(8\) m s\(^{-1}\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude \(20\) N.

1. Use the work-energy principle to find the distance \(AB\). [5]