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# Tsiolkovsky rocket equation - acceleration based

## Description

The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum.

The equation relates the delta-v (the maximum change of velocity of the rocket if no other external forces act) to the effective exhaust velocity and the initial and final mass of a rocket, or other reaction engine.

ΔV is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v is not usually the actual change in speed or velocity of the vehicle.

Imagine a rocket at rest in space with no forces exerted on it (Newton’s First Law of Motion). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a constant mass flow rate p (kg/s) and at exhaust velocity relative to the rocket ve (m/s). This creates a constant force propelling the rocket that is equal to p × ve. The mass of fuel the rocket initially has on board is equal to m0 – mf. The mass flow rate is defined as the total wet mass of the rocket over the combustion time of the rocket, so it will therefore take a time that is equal to (m0 – mf)/p to burn all this fuel. The rocket is subject to a constant force (M × ve), but its total weight is decreasing steadily because it is expelling gas. According to Newton’s Second Law of Motion, its acceleration at any time t is its propelling force divided by its current mass:

Related formulas

## Variables

 a acceleration (m/s2) p mass flow rate (kg/s) ve exhaust velocity relative to the rocket (m/s) m0 initial total mass (kg) t time (s)