by one and as quickly as possible, these stones in the opposite direction to the bank. Effectively, the quantity of movement of the stones thrown in one direction corresponds to an equal quantity of movement for the boat in the other direction. Derivation Most popular derivation Consider the following system: In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant". Newton's second law of motion relates external forces () to the change in linear momentum of the whole system (including rocket and exhaust) as follows: where is the momentum of the rocket at time : and is the momentum of the rocket and exhausted mass at time : and where, with respect to the observer: is the velocity of the rocket at time is the velocity of the rocket at time is the velocity of the mass added to the exhaust (and lost by the rocket) during time is the mass of the rocket at time is the mass of the rocket at time The velocity of the exhaust in the observer frame is related to the velocity of the exhaust in the rocket frame by (since exhaust velocity is in the negative direction) Solving this yields: and, using , since ejecting a positive results in a decrease in mass in time, If there are no external forces then (conservation of linear momentum) and Assuming that is constant (known as the Tsiolkovsky's hypothesis), so it is not subject to integration, then the above equation may be integrated as follows: This then yields or equivalently or or where is the initial total mass including propellant, the final mass, and the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration). If is NOT constant, we might not have rocket equations that are as simple as the above forms. Many rocket dynamics researches were based on the Tsiolkovsky's constant hypothesis. The value is the total working mass of propellant expended. (delta v) 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)