• Post category:Rocket Science
  • Reading time:5 mins read

Consider a rocket in space and an observer standing on earth.

Rocket Equation Derivation
Rocket Equation Derivation

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 + ΔmVeMV – 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.

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