Rocket equation is a straight-forward outcome, if we understand some core ideas and assimilate them.
Idea 1: Measurement and analysis of properties of body in motion is called Kinematics. Kinematic properties include displacement, velocity, acceleration, jerk etc. Momentum is a mechanical quantity which takes in account of the mass of a moving object along with its kinematic variable — velocity
Idea 2: Conservation of momentum
Net momentum of a closed system remains constant unless there is an external influence — momentum is conserved. Assume you are geared up with your roller skates and you throw a stone forward (imparting impulse), stone picks up velocity in forward direction. You will notice that you start to move in opposite direction to cancel out stone’s momentum, since the initial net momentum was zero (you and stone were at rest).
Rocket works almost similar to the above roller-skate and stone scenario. Rocket ejects a continuous flux of hot gases at very-high velocities via its nozzle as a result of combustion in its chambers. As you might have guessed, the rocket has to move in opposite direction (of exhaust gas) to conserve the momentum of the closed system, thus creating a flight!
To arrive at the governing equation of rockets, let consider a small interval of time (Δt), in which rocket ejects a finite mass (Δm) of gas at speed u, which results in rocket’s change in forward velocity (Δv).
Net momentum before and after the burn should be equal since there is no external influence. Equating the momentum yields our difference equation.
M — Mass of rocket excluding the ejected gas
V — Initial velocity of rocket
u — Exhaust gas velocity
Δm — Ejected mass
Δv — Change in rocket’s velocity (conserving the momentum)
Δt — Considered finite time
Note that exhaust gas speed u, is measured from the rocket’s frame of reference.
This resulting equation is the governing equation of rocket engine!
Taking integral over time yields equation for rocket’s velocity as a function of time and varying mass of rocket.
