dKE dt Note that we counted both vehicle and wake Ke as produced by E, and this is unambiguous. If we try to define useful Propulsive work"as Fv=mcv, then we find Fv mvc 2V E is arbitrarily high! IfV> For this reason, nprop is not used for rockets. But it is still true that thrusting at high △ +cIn +v.cIn In the presence of external air, some modification is needed, leading to the well Ju+Ae(P -Pa)=mc defines c=u=Jet speed far from exhaust For finite Par in thermal rockets, increasing Ae increases ue(towards a limit To), but it eventually makes(Pe-P)A The best Ae is such as to make pe The thrust coefficient c is used to quantify the performance of nozzles. Starting from =mu+Ae(P-P) 16.522, Space P pessan Lecture 1b Prof. Manuel martinez Page 2 of 616.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 2 of 6 ( ) th d KE = E, dt η i then 2 th 1 E= mc 2 η i i jet kinetic power Note that we counted both vehicle and wake KE as produced by E i , and this is unambiguous. If we try to define “useful Propulsive work” as Fv = m cv, i then we find that the “propulsive efficiency” prop 2 th F.v mvc 2v = = 1 c E mc 2 η = η i i i is arbitrarily high! pr. c If v > , > 1 . 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ η For this reason,ηprop is not used for rockets. But it is still true that thrusting at high speed increases kinetic energy more f 2 2 2 0 f0 0 f 1 1 m ∆ m v = m v + c ln - v 22 m ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ In the presence of external air, some modification is needed, leading to the well￾known formula ( ) F =mu +A P -P mc e ee a ≡ i i defines e c u  = Jet speed far from exhaust For finite Pa, in thermal rockets, increasing Ae increases ue (towards a limit e p0 max u 2c T ≅ ), but it eventually makes e ae (P - P )A negative. The best Ae is such as to make Pe = Pa. The thrust coefficient cF is used to quantify the performance of nozzles. Starting from ( ) F =mu +A P -P e ee a i (or finite Pe) 2 2 2 0 0 f 0 f f 1 m m = m c ln + v c ln 2m m ⎡ ⎤ ⎢ ⎥ ⎣ ⎦