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 **Δ**m 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 **Δ**V = 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.