Velocity and mass rate of a rocket.

[AbigailM]

Homework Statement

A rocket ejects pressurized air with constant relative velocity $v_{rel}$ and moves horizontally. Starting from rest and an initial mass $m_{1}$, find the speed of the rocket when its mass is $m_{2}(m_{2}<m_{1})$. How does this result depend on the rate $r=dm/dt$ at which the air is ejected?

Homework Equations

$v = v_{0}+v_{rel}ln\frac{m_{1}}{m_{2}}$

$m(t)=m_{1}+\dot{m}t$

The Attempt at a Solution

$v=v_{rel}ln\frac{m_{1}}{m_{2}}$ where $v_{0}=0m/s$

The result depends on the rate in that the velocity increases as $m_2$ becomes smaller.

How does this look? Thanks for the help.

 

[Filip Larsen]

Your answer to the first question is correct, but you may want to think some more about the second question. You should try to answer that question under the assumption that m₁, m₂ and v_rel all are fixed values.

 

[AbigailM]

Ok I think I've got something.

From the continuity equation for mass,
$\dot{m}=\rho v A$ where $\rho$ is the density and A is the area.

$\dot{m}=\rho Av_{rel}ln\frac{m_{1}}{m_{2}}$

Does this look ok?

Thanks!

 

[Filip Larsen]

No, it is simpler than you think.

Your original equation (the rocket equation) is correct, i.e. ΔV = V₂ - V₁ = V_e ln(m₁/m₂). Now, the question is if this equation depends on the value of the mass rate, that is, assuming that V_e, m₁ and m₂ all are constant do ΔV then change if dm/dt change, or in other words, do dm/dt appear in the rocket equation at all?

Looking at the rocket equation the answer is obvious, but the result is perhaps a bit surprising given how present dm/dt is in the derivation leading up to the rocket equation. It means, for instance, that a spacecraft will not end up going faster if you increase the accelerating force by putting on an extra (identical) rocket engine but keep the amount of propellant equal. The increased force will be "compensated" by the increased mass flow rate so that the rocket, while accelerating faster, will use its propellant faster and still end up with the same total change in speed. This means that change in speed for a given engine technology (V_e) and mass ratio (m₁/m₂) is independent of the "size" of the engine and the reasons why the concept of "delta-Vee" is such a useful concept in space mission design.