# Braking force required to allow bike to move down slope with an arbitrary constant velocity

Find the braking force required to allow a bike to move down a frictionless slope, inclined at $\theta$ radians above horizontal, with a constant velocity of v m/s.

I'm confused because Newton's 2nd Law of motion implies that the sum of forces on the bike must be zero in order for velocity to be constant, meaning that the braking force required is simply the bike's weight, resolved long the direction of the slope, in the opposite direction. But this only makes sure that velocity is constant.

How is it possible to apply a force to make the bike move with a constant velocity, of exactly v? My guess is that it has something to do with work, energy and power, but I haven't found a way yet.

• One has to wonder how brakes would operate on a frictionless slope where it makes no difference whether the brakes are on because the tyres can slide over the surface without slowing the bike. Jul 6, 2015 at 10:53
• They could be air brakes.
– bdsl
May 7, 2017 at 20:04

Draw a force diagram and determine the braking force required to counteract the (portion of) gravitational force pulling the bike down the slope. If the net force is zero, delta-v will be zero. I think the problem you were given assumes the bike starts out with velocity v . If it doesn't then you'll need to derive a braking force whose magnitude varies as a function of velocity in some manner that goes asymptotic at the desired final velocity.