Undergrad Multistage continuous Rocket Eqn

[syncphysics99]

So if you have a rocket let's say that discards all the structural and engine mass continuously at zero velocity that is relative to the rocket until only the payload is traveling at the final velocity - then what will the equation of motion will look like? we can neglect the drag and gravitational losses.

 

[russ_watters]

If the rocket ejects mass at zero velocity relative to itself, it doesn't accelerate.

Anyway, for more usable assumptions, plug them into the rocket equation:
https://en.m.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

 

[jbriggs444]

So if you have a rocket let's say that discards all the structural and engine mass continuously at zero velocity that is relative to the rocket until only the payload is traveling at the final velocity - then what will the equation of motion will look like? we can neglect the drag and gravitational losses.

You could make either of two assumptions.

1. The structure and engine mass are ejected in discrete chunks. i.e. stage 1 is burned out over time then the useless structure and engine are discarded instantly. In this case you can apply the Tsiolkovsky rocket equation piecewise over each burn.

2. The structure and engine mass are dropped continuously and in proportion to the burnt fuel/ejected reaction mass. In this case you can multiply the exhaust mass flow rate by the exhaust velocity to get a momentum ejection rate. Then divide by the combined (and proportional!) engine+superstructure ejection rate to get an "effective exhaust velocity". Plug that into the Tsiolkovsky rocket equation applied over the total burn.