# 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