Effective mass (solid-state physics) from both sides gives where is the th element of , is the th element of , and are the th and th elements of , respectively, and is the total energy of the particle according to the Planck–Einstein relation. The index is contracted by the use of Einstein notation (there is an implicit summation over ). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor This tensor expresses the change in group velocity due to a change in crystal momentum. Its
Variable-mass system on a body (F = 0) the variable-mass system motion equation reduces to If the velocity of the ejected propellant, v, is assumed have the opposite direction as the rocket's acceleration, dv/d"t", the scalar equivalent of this equation can be written as from which d"t" can be canceled out to give Integration by separation of variables gives By rearranging and letting Δ"v" = "v" - "v", one arrives at the standard form of the ideal rocket equation: where "m" is the initial total mass, including propellant, "m" is the final total mass, "v" is the effective exhaust velocity (often denoted