Rocket Velocity Determination with Constant Mass Loss and Resistive Force

[ElderBirk]

Homework Statement

Consider a rocket is subject to linear resistive force, $$f = -bv$$. $$\dot m$$ is constant. Use the equation: $$m \dot{v} = -\dot{m }v + f$$ to determine the velocity of the rocket :

since the rate of mass lost is constant
let $$\dot{m} =k$$
vex = nuzzle velocity

$$v = \frac{k}{b} vex (1 - (\frac{m}{m_0})^{\frac{b}{k}})$$

Homework Equations

Already given above

The Attempt at a Solution

let $$m= \frac{dm}{dt} \cdot dt = k \cdot dt$$

$$k dt \frac{dv}{dt} = -kvex - bv$$
I don't know if cancelling $$dt$$'s are allowed but when I solve this equation, it doesn't resemble the expected answer at all.

 

[zhermes]

You say 'm' is constant... but you have a non-zero m derivative.
Also, $$m = \int \frac{dm}{dt} dt$$

 

[ehild]

Make it clear if k is the rate of loss of mass, which means dm/dt=-k or k=dm/dt.
m means the mass of the racket at time t. You need m(t), the expression of mass in terms of time. The original mass is m₀. The loss of mass is constant, k. What is m(t), the mass t time after start? It is certainly not kdt as you wrote. ehild

 

[ElderBirk]

I believe I figured it out. Sorry for the confusion. dm/dt =k.

for future reference, if one is looking at this problem, they can easily solve it by raping calculus, i.e.

m dv/dt = m dm/dm dv/dt = m dm/dt dv/dm = m k dv/dm

Now the only dependencies are on m and v, thus by separation of variables the problem becomes solvable.