# Need help creating an equation for the time it takes for a rocket to deaccelerate (accounting for fuel mass lost)

I'm doing a Math Internal Assesment for school where I'm trying to create a model for a rocket's suicide burn. I've gotten stuck in one of the equations I need for it. I need to find the time it would take me to deaccelerate the rocket. This WOULD be a simple kinematics equation (I might have made a mistake with the negative, but w/e for now):

$t = \frac{-u}{a}$ or $t = \frac{-u}{-gravity+\frac{Force}{mass}}$

This is fine if the mass of the rocket was constant. However, if I want to also model how acceleration changes as fuel mass goes down, things get ugly. The rate of mass change is simply $m - (f * t)$ where m is initial (wet) mass, f is the rate of fuel mass consumed per second and t is time. This makes sense, right? However when I tried to plug this in I ended up with this equation:

$$t=\frac{-u}{-gravity+\frac{F}{m - (f * t)}}$$

FWIW the units for my situation are: gravity is 1.69m/s, initial velocity (u) would be variable, F is 60,000 Newtons, initial mass is 3840 kg and the rate of fuel consumption is 17.71kg/s.

Look at the equation, it's preposterous. t is on both sides. t is being defined partly by itself. Yet I can't wrap my head around any alternative. I need to find an answer urgently.

I'm thinking there has to be a way for the rate of mass change to be affected by time's magnitude, but there's no math in my knowledge that would allow such a thing to be done, so I turn to you guys.

• – Emilio Pisanty Sep 20 '18 at 2:57
• Seriously, though: is the mass change from the suicide burn actually a significant part of the rocket's mass at landing in realistic models? – Emilio Pisanty Sep 20 '18 at 3:00

## 1 Answer

Basically, you need to re-formulate and solve a version of the Tsiolkovsky rocket equation, if you want to fully account for the effect of the mass loss. Basically, you need to find an appropriate formulation of the rocket's equation of motion, using mass-loss terms similar to the lifting-rocket versions, and then you solve that. Since the ODE's structure gets significantly altered, the solution will look quite different to the no-mass-loss version you started with.

If that looks too complicated, though, I would point out that with the existing rockets that can perform them, the landing burn is performed close to the end of the fuel lifetime (so the mass is mostly dry mass) and the fuel expenditure is (nontrivial but) relatively small, so its effects on the kinematics will be much smaller than they are during the lifting phase. As such, I would suggest researching realistic models and seeing whether the modification of the EOM actually adds any important accuracy.