5. 522, Space Propulsic Prof. Manuel martinez-sanchez Lecture 1b: Review of Rocket Propulsion The only practical way to accelerate something in free space is by reaction. The idea is the same as in air breathing propulsion(to push something backwards)but in ockets the something"must be inside and is lost Here is a revealing derivation of the thrust equation for vacuum v(t) mt Mom()=M()v()+m(t)v()-c(e=constant M(t clt no drag, no interaction f molecules with ambient air d(Mom) 0 dM+ m(v-c) But dM.dv m=- mc call this thrust , f. Notice d(Mv) dt =F-vm=m(c-v) which can be (+)or(-) Using the same technique, the kinetic energy of the system rocket-jet is KE=SMV+5m(t[v(t)-c(t] dKE dv dt dM+1m(+c2-2vxc)=2mce dt 2 mcv So, if thermal (or electrical) energy is expended internally at the rate E, and converted to total kinetic energy with efficiency nth(or ne,) 16.522, Space P pessan Lecture 1b Prof. Manuel martinez Page 1 of 616.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 1 of 6 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 1b: Review of Rocket Propulsion The only practical way to accelerate something in free space is by reaction. The idea is the same as in air breathing propulsion (to push something backwards) but in rockets the “something” must be inside and is lost. Here is a revealing derivation of the thrust equation for vacuum: () () () ( ) ( ) ( ) t 0 Mom. t = M t v t + m t' v t' - c t' dt' = Constant ⎡ ⎤ ∫ ⎣ ⎦ i d Mom ( ) = 0 dt ( ) dv dM M +v +m v-c =0 dt dt i But dM dv m=- M =mc dt dt → i i call this thrust, F. Notice ( ) ( ) d Mv =F-vm=m c-v dt i i which can be (+) or (-) Using the same technique, the kinetic energy of the system rocket-jet is () () () t 2 2 0 1 1 KE = Mv + m t' v t' - c t' dt' 2 2 ⎡ ⎤ ∫ ⎣ ⎦ i So, ( ) dKE dv 1 dM 1 1 2 22 2 = Mv + v m v + c - 2vc m c dt dt 2 dt 2 2 + = i i m cv i −m i So, if thermal (or electrical) energy is expended internally at the rate E, i and converted to total kinetic energy with efficiency th e1. η (or ) η ; i.e., no drag, no interaction of molecules with ambient air