# 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.

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