11: Everything You Need to Know about the Rocket Equation (Physics and Applied Mathematics)

What is the Rocket Equation?

The rocket equation is a fundamental principle in physics, based on Newton’s Third Law of Motion) that describes the motion of rockets in space. It is a mathematical formula that calculates the velocity of a rocket based on its mass, the mass of the propellant it carries, and the speed at which it expels that propellant.

Credit: NASA

 

The equation shows that as the mass of the rocket decreases (as it expels propellant), its velocity increases. This is because the rocket's momentum is conserved, and as the mass decreases, the velocity must increase to maintain momentum.

The rocket equation is important because it determines the amount of propellant required to achieve a certain velocity or distance in space. It also shows that there are limits to the performance of rockets, as increasing the propellant mass requires a corresponding increase in the initial mass of the rocket, which makes it heavier and more difficult to launch.

Credit: The Planetary Society

 What's the Physics behind this? (Calculus-I prerequisite)

Here, we derive this equation:

Take the initial momentum of the rocket to be the mass of the rocket (m) multiplied by the velocity (V).

For the final momentum, there is a loss of dm (relatively low) mass of propellant expelled, the rocket’s mass becomes m-dm. The propellant moves at a velocity of V-Velocity of exhaust (Ve) and the rocket moves at a velocity of V + dv (small change)

This gives us dm(V-Ve) + (m-dm)(V+dv)

Initial momentum = final momentum, hence

mV = dm(V-Ve) + (m-dm)(V+dv)

mV = dmV – dmVe + mV + mdu – dmV – dmdv

 

dmdv is far too small, so it can approximated and removed from the equation. 

mV cancels on both sides and dmV cancels -dmV, leaving us with:

 

-dmVe + mdv  = 0

Hence, mdv = Vedm

 

Multiplying by 1/m on both sides, we get

dv = Ve * dm * 1/m

 

Integrating both sides,

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Hence,

 

[V]^Vf _V0  = Ve [ ln(m)]^mf _m0 

 

Which yields:

 

Vf = Ve {ln(mf) – ln(m0)}

Hence, Vf = Ve {ln (mf/m0)}

So Vf/Ve = ln (mf/mo) 

This formula then yields the ratio of final payload (mf) to total initial mass (m0) in terms of a ratio of velocity.

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