Work, Energy and Power question

[gaobo9109]

Homework Statement

A cyclist develops constant power when cycling. The air resistance is proportional to the speed of cycling. When he cycles on level ground, the maximum velocity he can attain is v. When he cycles up a slope, his maximum velocity is 1/2v. If he cycles down the same slope, his maximum velocity is av. Ask what is the value of a.

Homework Equations

The Attempt at a Solution

The cyclist exerts the same amount of force on the bike at all velocity,thus

On level ground: F = kv₂

cycling up a slope: F = 1/4kv² + mgsinθ
kv² = 1/4kv² + mgsinθ
mgsinθ = 3/4kv²

cycling down the slope: kv² + 3/4kv² = k(av)²
a² = 7/4
a = (7/4)^1/2

But this is not the correct answer. Can anyone tell me where did I go wrong?

 

[PhanthomJay]

You assumed that the air resistance was proportional to the square of the speed (kv^2), when it is given that it is proportional to the speed (kv).