Transcendental Retarding Force

[bndnchrs]

Homework Statement

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

Homework Equations

F=ma

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).

 

[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!

 

[Rainbow Child]

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