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.

 

[paranormal]

I would approach this problem by first clarifying the variables and their meanings. In this case, the rocket is subject to a linear resistive force, represented by the variable f. The rate of mass loss is represented by \dot{m} and is assumed to be constant, represented by the constant k. The mass of the rocket is represented by m and its initial mass is represented by m_0. The velocity of the rocket is represented by v and the nuzzle velocity is represented by vex.

Next, I would analyze the given equation, m \dot{v} = -\dot{m }v + f, and determine what it represents. This equation is a representation of Newton's Second Law, which states that the net force acting on an object is equal to its mass times its acceleration. In this case, the net force is the sum of the linear resistive force and the force due to the mass loss of the rocket.

To determine the velocity of the rocket, I would rearrange the equation to solve for v. This would give me v = \frac{1}{m}\left(-\dot{m}v + f\right). Since \dot{m} is constant, we can substitute in its value of k, giving us v = \frac{1}{m}\left(-kv + f\right). Next, I would substitute in the given value of f = -bv, giving us v = \frac{1}{m}\left(-kv - bv\right). Finally, I would rearrange the equation to solve for v, giving us v = \frac{k}{b}vex\left(1 - \left(\frac{m}{m_0}\right)^{\frac{b}{k}}\right).

This final equation represents the velocity of the rocket as a function of time and the variables given. It shows that the velocity of the rocket decreases as its mass decreases, and it approaches the nuzzle velocity as the mass approaches zero. This equation can be used to determine the velocity of the rocket at any given time, as long as the values of m, m_0, b, k, and vex are known.