# Acceleration function of velocity example

## Homework Statement

Because the drag on objects moving through air increases as the square
of the velocity, the acceleration of a bicyclist coasting down a slight hill
is (v) = a - cv where a and c are constant. Determine the velocity of
the bicyclist as a function of distance if the velocity is zero when x=0.
Also determine the maximum velocity that the cyclist attains.

dx/dt = v
dv/dt = a-cv

## The Attempt at a Solution

x = -v/c - (a/c^2)*ln(a-cv)/a
v = (a - ae^-ct)/c
t = -(cx + v)/a = -(1/c)*ln(a-cv)/a

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"...the acceleration of a bicyclist coasting down a slight hill
is (v) = a - cv where a and c are constant."

Shouldn't the v on LHS be dv/dt?

Hint for v as a function of distance:

V = dx/dt = (dx/dv)*(dv/dt) = (dx/dv)*(a - c*V)

Maximum velocity occurs when acceleration is zero. which is what your second equation shows for large t.