m(t)c(t)=5F(t)c(t) Consider the rate of change of the inverse mass with time 1】) dM m (16a) Multiplying and dividing by F2=m c (16b) M2 mc2 where a= F/M is the acceleration due to thrust Inte 111 (17) P On the other hand, the mission Av is △V=adt (18) and is a prescribed quantity. We wish to select the function a(t) which will give a naximum M(Equation 17)while preserving this value of Av. The problem reduces to finding the shape of a(t), whose square integrates to a minimum while its own value has a fixed integral. the solution(which can be found by various mathematical techniques but is intuitively clear) is that a should be a constant Using this condition, (17)and(18)integrate immediately. Eliminating a betweer these, we obtain M 1 (19) M M△V ntp The level of power is yet to be selected; it will determine the average specific impulse, and it is to be expected that an optimum will also exist U M=M+M d 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 7 of 1916.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 7 of 19 () () () () 1 1 2 P= m t c t = F t c t 2 2 η η i (15) Consider the rate of change of the inverse mass with time: 2 2 1 d M 1 dM m =- = dt dt M M ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ i (16a) Multiplying and dividing by 2 2 2 F =m c , i 2 2 2 2 1 d M F a = = dt 2 P M mc ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ η i (16b) where a=F M is the acceleration due to thrust. Integrating, t 2 f 0 0 11 1 - = a dt M M 2Pη ∫ (17) On the other hand, the mission ∆V is t 0 ∆V = a dt ∫ (18) and is a prescribed quantity. We wish to select the function a(t) which will give a maximum Mf (Equation 17) while preserving this value of ∆V . The problem reduces to finding the shape of a(t), whose square integrates to a minimum while its own value has a fixed integral. The solution (which can be found by various mathematical techniques, but is intuitively clear) is that a should be a constant. Using this condition, (17) and (18) integrate immediately. Eliminating a between these, we obtain f 2 0 0 M 1 = M M V 1+ 2 tP ∆ η (19) The level of power is yet to be selected; it will determine the average specific impulse, and it is to be expected that an optimum will also exist. Using M =M +M f Ls and