Question

The differential equation for an object coasting to a stop is dv/dt = -kv, where k is constant.

Claim : The equation that gives s(t) for an object coasting to a stop is
s(t) = (initial v)/ k = (1- e^-kt)

Here, s(t) represents the distance traveled while coasting.
Is the claim true or false?

Answer 1

well, do the math.
dv/dt = -kv
dv/v = -k dt
ln v = -kt+C
v = e^(-kt+C) = c*e^(-kt)
at t=0, v0 = c
So, v(t) = v0 * e^(-kt)
Now, s(t) = -1/k * v0 * e^(-kt) + C = -v0/k e^(-kt) + C
at t=0 we assume s=0, -v0/k = -C
and s(t) = -v0/k (1 - e^(-kt))