Question

The power output of a cyclist moving at a constant speed of $6.0 \mathrm{m} / \mathrm{s}\( on a level road is \)120 \mathrm{W} .$ (a) What is the force exerted on the cyclist and the bicycle by the air? (b) By bending low over the handlebars, the cyclist reduces the air resistance to \(18 \mathrm{N} .\) If she maintains a power output of \(120 \mathrm{W},\) what will her speed be?

Short answer

(b) If the cyclist bends low over the handlebars and reduces air resistance to 18 N, what is her new speed while still maintaining the same power output? Answer: (a) The force exerted on the cyclist and bicycle by the air is 20 N. (b) When the cyclist reduces the air resistance to 18 N and maintains a power output of 120 W, her speed will be 6.67 m/s.

Step-by-step solution

Identify the given information and the equations to use

The given information includes the cyclist's constant speed (\(v=6.0 \thinspace m/s\)) and power output (\(P = 120 \thinspace W\)). We need to find the force exerted by the air in part (a) and then the new speed in part (b) when the air resistance is reduced. We will use the power equation: \(P=F \cdot v\), where \(P\) is the power, \(F\) is the force, and \(v\) is the speed.

Solve for the force exerted by the air in part (a)

Using the power equation and the given values, we have: \(120 = F \cdot 6.0\) Now, we can solve for the force \(F\): \(F = \frac{120}{6.0}\) \(F = 20 \thinspace N\) The force exerted on the cyclist and the bicycle by the air is \(20 \thinspace N\).

Solve for the new speed when air resistance is reduced in part (b)

As given, when the cyclist bends low over the handlebars, the air resistance is reduced to \(18 \thinspace N\). Since the power output remains the same (\(120 \thinspace W\)), we can use the power equation with the new force value to find the new speed: \(120 = 18 \cdot v\) Now, we can solve for the new speed \(v\): \(v = \frac{120}{18}\) \(v = 6.67 \thinspace m/s\) When the cyclist reduces the air resistance to \(18 \thinspace N\) and maintains a power output of \(120 \thinspace W\), her speed will be \(6.67 \thinspace m/s\).