and introducing the characteristic velocity(Equation 6), we rewrite(19)as 1 (20) MM[(M,M5)+(△V/ M and select the value of s that will maximize L. This is easily found to be (21) hich, when used back in(21) gives and then These are, within the assumptions, exact expressions. they are to be compared to the approximate expressions in Figure 2 or Equation(12)-(14)with 8=0, 50=0 which were found to apply when C, and not a, was assumed constant. Clearly, the difference is noticeable only for v= Av/vch near unity(its highest value), and is negligible for smaller values. It is of some interest to inquire at this point how the jet velocity c should vary with time in order to keep the acceleration constant. We have mc m MM where(16 a, b)have been used. Hence, c=2n1 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 8 of 1916.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 8 of 19 Ms P = α and introducing the characteristic velocity (Equation 6), we rewrite (19) as ( ) ( ) L s 2 0 0 s 0 ch M M 1 = -1 M M M M + Vv ⎡ ⎤ ⎢ ⎥ ∆ ⎣ ⎦ (20) and select the value of s 0 M M that will maximize L 0 M M . This is easily found to be s 0 ch ch OPT M V V = 1- M vv ⎛⎞ ⎛ ⎞ ∆ ∆ ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ (21) which, when used back in (21) gives 2 L 0 ch MAX M V = 1- M v ⎛⎞ ⎛ ⎞ ∆ ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ (22) and then P 0 ch OPT M V = M v ⎛ ⎞ ∆ ⎜ ⎟ ⎝ ⎠ (23) These are, within the assumptions, exact expressions. They are to be compared to the approximate expressions in Figure 2 or Equation (12)-(14) with s0 0 M = 0, = 0 M δ which were found to apply when c, and not a, was assumed constant. Clearly, the difference is noticeable only for v= V v ∆ ch near unity (its highest value), and is negligible for smaller values. It is of some interest to inquire at this point how the jet velocity c should vary with time in order to keep the acceleration constant. We have 2 2 mc m a a = = Mc = Mc M 2P M η i i where (16 a, b) have been used. Hence, 2 P1 c = a M η