Consider a rocket in space and an observer standing on earth.
At time t=0, the rocket’s total mass is M + Δm. Where M is the mass of the empty rocket and Δm is the mass of the fuel. The entire system is moving at a velocity of V, with respect to an observer on earth.
Total Initial momentum of the rocket = Mass x Velocity
Pi = (M + Δm) x V
At time t= T, the rocket’s engine burns and ejects the fuel, and gains velocity.
The rocket’s velocity becomes V + ΔV and the rocket’s exhaust velocity becomes Ve. Both velocities are with respect to an observer on earth.
Total final momentum of the rocket = Mass of rocket x Rocket’s velocity + Mass of Exhaust x exhaust’s velocity.
Pf = M x (V + ΔV) + Δm x Ve
Since there is no external force acting on the rocket, the net force is zero. Hence, the total change in momentum from t=0 to t=T will be zero.
Pf – Pi = 0
M x (V + ΔV) + Δm x Ve – (M + Δm) x V = 0
MV + MΔV + ΔmVe – MV – ΔmV = 0
MΔV + ΔmVe – ΔmV = 0
Now, Ve and V are the velocity of exhaust and rocket, respectively, with respect to an observer on earth.
The relative velocity of exhaust with respect to the rocket is —
u = V – Ve
or Ve = V – u
Adding that in the above equation we get —
MΔV + Δm(V – u) – ΔmV = 0
MΔV + ΔmV – Δmu – ΔmV = 0
or MΔV – Δmu = 0
Now, understanding the fact that Δm results in a decrease in the total mass of the rocket system.
Hence Δm = – dm
And further considering an infinitesimal time difference dt from time t=0 to t=T, we have ΔV = dv
So the final equation becomes.
MdV = – udm
or dV = – u dm/M
Now integrating it over the small-time interval dt, from the final state to initial state and noting that integral of dx/X = ln X, we get —
Vf – Vi = – u {ln (Mf) – ln (Mi)}
Vf – Vi= u {ln (Mi) – ln (Mf)}
Vf – Vi = u ln (Mi / Mf)
This is the Tsiolkovsky Rocket Equation
Vf – Vi denotes “Delta-v”, which is the impulse per unit mass required for any maneuver, for example, change of rocket’s orbit from one radius to another radius.
u is the Effective exhaust velocity
and Mi / Mf represents the ratio of the “Total” rocket’s initial and final mass.