mass is ejected at speed and the remaining mass of the rocket is , the amount of energy converted to increase the rocket's and pellet's kinetic energy is Using momentum conservation in the rocket's frame just prior to ejection, , from which we find Let be the initial fuel mass fraction on board and the initial fueled-up mass of the rocket. Divide the total mass of fuel into discrete pellets each of mass . The remaining mass of the rocket after ejecting pellets is then . The overall speed change after ejecting pellets is the sum Notice that for large the last term in the denominator and can be neglected to give where and . As this Riemann sum becomes the definite integral since the final remaining mass of the rocket is . Special relativity If special relativity is taken into account, the following equation can be derived for a relativistic rocket, with again standing for the rocket's final velocity (after expelling all its reaction mass and being reduced to a rest mass of ) in the inertial frame of reference where the rocket started at rest (with the rest mass including fuel being initially), and standing for the speed of light in vacuum: Writing as allows this equation to be rearranged as Then, using the identity (here "exp" denotes the exponential function; see also Natural logarithm as well as the "power" identity at Logarithmic identities) and the identity (see Hyperbolic function), this is equivalent to Terms of the equation Delta-v Delta-v (literally "change in velocity"), symbolised as Δv and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse that is needed to perform a maneuver such as launching from, or landing on a planet or moon, or an in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle. Delta-v is produced by reaction engines, such as rocket engines and is proportional to the thrust per unit mass, and burn time, and is used to determine the mass of propellant required for the given manoeuvre through the rocket equation. For multiple manoeuvres, delta-v sums linearly. For interplanetary missions delta-v is often plotted on a porkchop plot which displays the required mission delta-v as a function of launch date. Mass fraction In aerospace engineering,