How can I find the time required to achieve certain velocity in orbit with changing mass?

Question

For example how long would it take for Atlas V 541 Stage two to accelerate to 7800m/s given it is travelling at 5000m/s. The mass of the rocket is changing over time, how do I find the time required for it to achieve 7800m/s?

Answer 1

What you want to do is to find how much propellant mass that velocity change is going to cost, and divide that by the engine mass flow to get the elapsed time.

Mass flow is easy, that is just:

$$\frac{Thrust}{I_{sp} \cdot g}$$

But how can we find the propellant consumption? Consider the rocket equation:

$$\Delta v = \ln \left({\frac{m_0}{m_f}}\right)\cdot I_{sp} \cdot g$$

We can use that to get the final mass:

$$m_f = \frac{m_0}{e^{\frac{\Delta v}{I_{sp} \cdot g}}}$$

Propellant consumption is then $m_0 -m_f$

Here is the combined equation for convenience:

$$Time = \frac{I_{sp} \cdot g \cdot m_0\left(1 - \frac{1}{e^{\frac{\Delta v}{I_{sp} \cdot g}}}\right)}{Thrust}$$

Alternatively, we can formulate it in terms of $m_f$ if we only have the final mass:

$$Time = \frac{I_{sp} \cdot g \cdot m_f\left(e^{\frac{\Delta v}{I_{sp} \cdot g}}-1\right)}{Thrust}$$