How does burn time affect the parameters in the rocket equation?

[zenterix]

Homework Statement

a) Before a rocket begins to burn fuel, the rocket has a mass of $m_{r,i}=2.81\times 10^7\mathrm{kg}$, of which the mass of fuel is $m_{f,i}=2.46\times 10^7\mathrm{kg}$. The fuel is burned at a constant rate with total burn time $\mathrm{510s}$ and ejected at a speed of $u=3000\mathrm{m/s}$ relative to the rocket. If the rocket starts from rest in empty space, what is the final speed of the rocket after all the fuel has been burned?

b) Now suppose the same rocket burns the fuel in two stages ejecting the fuel in each stage with the same relative speed. In stage one, the available fuel to burn is $m_{f,1,i}=2.03\times 10^7\mathrm{kg}$ with burn time $\mathrm{150s}$. Then the empty fuel tank and accessories from stage one are disconnected from the rest of the rocket. These disconnected parts have a mass of $1.4\times 10^6\mathrm{kg}$. All the remaining fuel is burned during the second stage with a burn time of $\mathrm{360s}$. What is the final speed of the rocket after all the fuel has been burned?

Relevant Equations

$\vec{F}_{ext}=m_r(t)\vec{v}_r'(t)-um_r'(t)$

The items (a) and (b) are provided for context. I am not concerned with solving the problem. That is relatively easy.

My question is about the burn time. It doesn't seem to matter for solving the problem as it has been posed. All we care about is the states at the beginning an end of each stage.

I noticed that in (b) the first stage burns most of the fuel in a way shorter time compared to the second stage. I can see how this is realistic in some cases: a rocket has to leave the atmosphere first and this requires the most power and fuel. But the problem above takes place in empty space.

But doesn't this first stage scenario affect the ejection speed of the fuel in any way?

In general, how does the burn speed affect the problem?

 

[Hill]

It determines how long the ejection lasts.

 

[Filip Larsen]

In general, how does the burn speed affect the problem?

As you suspect, varying the burn time does not change the final speed change you get from the rocket equation as long as the ejection speed is kept the same (i.e. same rocket technology) and the initial and final masses are the same, as is explicitly stated in this problem.

In some practical applications however, like for a launcher that has to burn to lift a payload mass from ground to low Earth orbit, the exact acceleration profile matters a lot mostly due a combination atmospheric and gravitational losses, so for those kinds of problems the burn time (i.e. the acceleration) do indeed factor into how much effective speed change the rocket can provide.