Rocket Equation: Derivation

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

P_i = (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 V_e. 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.

P_f = M x (V + ΔV) + Δm x V_e

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.

P_f – P_i = 0

M x (V + ΔV) + Δm x V_e – (M + Δm) x V = 0

MV + MΔV + ΔmV_e – MV – ΔmV = 0

MΔV + ΔmV_e – ΔmV = 0

Now, V_e 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 – V_e

or V_e = 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 —

V_f – V_i = – u {ln (M_f) – ln (M_i)}

V_f – V_i= u {ln (M_i) – ln (M_f)}

V_f – V_i = u ln (M_i / M_f)

This is the Tsiolkovsky Rocket Equation

V_f – V_i 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 M_i / M_f represents the ratio of the “Total” rocket’s initial and final mass.