
air resistance
A man on a bicycle, of total mass 100 kg, is freewheeling at a constant speed of 15ms^1 down a hill with a gradient 10% (i.e. sin^1(0.10)). He wants to slow down to a safer speed, so he applies the brake lightly to produce a constant braking force of 84 N. The air resistance is proportional to the square of the speed.
a. Calculate the deceleration when he first applies the brake.
Several other questions follow.
Anyway the problem is the air resistance: in this model I know v^2 but I don’t know the constant k as in kv^2! I know how to solve these problems with k but without it I’m lost. Is there a way to work around the air resistance, i.e. without using k?

You can get the $\displaystyle k$ of $\displaystyle \vec{F}_{ar}=k\,v^{2}$ if you apply Newton's Second Law before the biker brakes. What are the forces on the biker before he brakes, and what is the acceleration?
Incidentally, I would think that a gradient of $\displaystyle 10\%$ would imply that the angle of inclination of the hill is given by $\displaystyle \tan^{1}(0.10),$ not $\displaystyle \sin^{1}(0.10).$
I don't think there's a way to get an exact numerical answer without finding $\displaystyle k$, but as I've said, I think there's a way to find $\displaystyle k$.
Can you see your way forward now?