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

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?

• From the 5000 m/s point, thrust is approximately constant; initial and final mass can be estimated from information on Spaceflight101 and changes approximately linearly with time. The rest is straightforward calculus, but I'm not gonna do it. – Russell Borogove Mar 3 '17 at 21:49
• Yes, but you need it to solve the problem, as @Hohmannfan shows below. – Organic Marble Mar 3 '17 at 23:55

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:

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

• In some applications you may not know $m_0$; how ugly is it to reformulate in terms of $m_f$? – Russell Borogove Mar 3 '17 at 22:29
• @RussellBorogove Slightly less, in fact. Added. – SE - stop firing the good guys Mar 3 '17 at 22:51
• That's very nice! I may find use for that in my launch simulator. Do you rearrange equations longhand on paper or do you have some software tools? – Russell Borogove Mar 4 '17 at 0:19