<0< We still have(Equation 1) d i sin a cos e but now(using a=1) (sin a cos e)=+sin a 2 cos ede==sin a s de Similarly we still have dR 2FR cos夏 which needs no averaging Dividing(33) by(32)and dropping the averaging sign d i tan a We also have d△V_F/M dt F cOS d△V1√/R dr 2 cos a 16.522, Space P pessan Lecture 3 Prof. Manuel martinez Page 5 of 916.522, Space Propulsion Lecture 3 Prof. Manuel Martinez-Sanchez Page 5 of 9 So, ( ) + for - < < 2 2 = 3 - for < < 2 2 ⎧ π π α θ ⎪⎪ α θ ⎨ π π ⎪ α θ ⎪⎩ (31) We still have (Equation 1) 2 di F R = sin cos , d M α θ θ µ but now (using α ≡ 1 ) 2 -2 1 2 sin cos = sin cos d = sin π α θ α θ θ α θ π π π ∫ 2 di 2 R F = sin d M α θ πµ ∴ (32) Similarly, we still have 3 dR 2FR = cos d M α θ µ (33) which needs no averaging. Dividing (33) by (32) and dropping the averaging sign, di tan = dR R α π (34) We also have d V FM = , dR dR dt ∆ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ and now 3 2 1 2 dR F R 2 cos , dt M = α µ 3 dV 1 R = dR 2 cos ∆ µ α (35)