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16 Chapter 2 The thrust on the spacecraft is equal and opposite to the time rate of change of the momentum of the propellant, which is the exhaust velocity of the propellant times the time rate of change of the propellant mass: d dmp T = dt (mpvex )= vex dt , (2.1-2) where mp is the propellant mass on the spacecraft and vex is the propellant exhaust velocity in the spacecraft frame of reference. The total mass of the spacecraft at any time is the delivered mass, md , plus the propellant mass: M(t)=md +mp. (2.1-3) The mass of the spacecraft changes due to consumption of the propellant, so the time rate of change of the total mass is dM = dmp . (2.1-4) dt dt which can be written as spacecraft initial velocity, vi , to the final velocity, v f , during which the mass changes from its initial value, md + mp , to its final delivered mass, md : Substituting Eq. (2.1-4) into Eq. (2.1-2) and equating with Eq. (2.1-1) gives Mdv= v dM, (2.1-5) vi md+mp M The solution to Eq. (2.1-7) is dt exdt dv= v dM. (2.1-6) ex M For motion in a straight line, this equation is solved by integrating from the v mdM f dv= vex d . (2.1-7)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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