# Transcendental Retarding Force

1. Jan 27, 2008

### bndnchrs

1. The problem statement, all variables and given/known data
A boat with initial speed v[o] is launched, and experiences a retarding force of F = -ae^Bv, where a=alpha=constant and b=beta=constant

Find v(t)
Find Time and Distance for the boat to stop

2. Relevant equations
F=ma

3. The attempt at a solution
the second part for tmie and distance, I am attempting to solve using mathcad with a perturbative approach incrementing vt o find dv/dt, reinserting v-dv/dt until dv/dt approaches closely to zero. This makes sense, however, only with known values for alpha and beta. I can't really think of a way to formulate v(t).

2. Jan 27, 2008

### bndnchrs

I retried the problem, and was able to do some integration by treating it as a differentials and deriving a seperable ODE. However, my answers seem very odd. Here's what I did.

m*dv/dt = -$$\alpha$$*e^$$\beta$$*v

dv$$/$$e^$$\beta$$*v = -$$\alpha$$*dt$$/$$m

letting u = e^-Bv
du = -B*e^-Bv

$$\int$$1/-B * du = $$\int$$-A/m dt

going through all the integration...

we get

v(t)=ln((A*B/m)*(t+constant))/-B

with v(0) = v0

constant = (m/a*B)*e^-Bv0

so... let && = A*B/m

v(t) = ln(&&*t + e^(-B*v0))/-b

which is weird to me, because the higher we set the initial velocity, the slower the speed is after a set t. Someone help!

3. Jan 27, 2008

### Rainbow Child

It isn't wierd, because $$\beta$$ must be negative, or else the boat would move with negative velocity!