# Bike Race

**Classical Mechanics**Level 4

Alice and Carla both want to race their brand new bikes. Carla is a little older and stronger than Alice, so they propose a deal. Alice can ride her bike down a hill while Carla has to ride on flat ground, but in return Carla gets to pick the length of the race. Let \(L\) be the length of the race. What is the largest integer value of \(L\) that Carla can choose such that she still wins the race?

**Details and Assumptions**:

Alice is riding down an incline of \(45^\circ\) off the ground

If you draw the hill as a right triangle, \(L\) corresponds to the base of the triangle.

Both Alice and Carla start from a dead stop, Carla accelerates at \(10 \text{ m/s}^2\) until she reaches \(20 \text{ m/s}\) (then she rides at a constant speed), and Alice just rides down the hill using only gravity to propel her.

Assume that gravity is \(10 \text{ m/s}^2\).

Ignore the effects of wind resistance and terminal velocity.