The Rocket Problem. Figure 1 Figure 2. r = (1) dt. P t. r r r. ext

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1 The Rocket Poblem In the ocket poblem, the goal is to set up the diffeential equation of motion fo the elocity of the ocket as a function of time. Hee is a paaphase of H&R s deiation. [Physics, Pats & 2, 3 d Ed, Daid Halliday and Robet Resnick, John Wiley & Sons, ew Yok, 978, p. 75.] Figue Figue 2 We stat with = () which we can wite in appoximate fom as F P = P f P i (2) fo the finite time inteal. In this expession, P is the (final) system momentum depicted in figue 2, and P i is the (initial) system momentum depicted in figue. Fom these figues we see that P f = ( M M )( + ) + Mu (3) and = M. (4) P i f Putting equations 3 and 4 into equation 2 we hae [( M M )( + ) + Mu] [ M] = (5) which we ewite as M = M + [ u ( + )]. (6)

2 ow, if we let appoach zeo, the configuation of figue 2 appoaches that of figue ; d that is, appoaches, the acceleation of the body in figue. The quantity M is the mass ejected in : this leads to a decease in the mass M of the oiginal body. Since dm/, the change in mass of the body with time, is intinsically negatie in this case, the positie quantity M/ is eplaced by dm/ as appoaches zeo. Finally, goes to zeo as appoaches zeo. Making these changes in equation 6 we hae d dm dm d dm = M + u = ( M) u (7) which is ewton s second law, defining the enal foces on a body (like that of figue ) whose mass is changing. It is impotant to note that we cannot deie a geneal expession fo ewton s second d( M) law fo aiable mass systems by teating the mass in = = as a aiable. Fo this leads to d d dm = ( M) = M +, (8) which is only a special case of the moe geneal equation 7 We can use = to analyze aiable mass systems only if we apply it to an entie system of constant mass haing pats among which thee is an intechange of mass. This is indeed what we hae done in deiing equation 7. H&R say they ae using = fo a system in which mass is constant, but M does not include expended fuel. In othe wods, each succeeding time inteal sees a new alue fo M in figue. So, clealy they ae dealing with a time-dependent mass. Thei comments ae claified/qualified in the following, wheein we stat by deiing =. Fo the entie uniese, momentum is conseed. Of all the substance in the uniese, we want to find the equation of motion fo a pope subset of mass M. Suppose we beak the uniese into subsets each with momentum P i, i =,, whee the i = subset is that of M. This gies PUni = Pi = Const. (9)

3 The time ate of change of equation 9 is Uni i = If we isolate ou subset on the LHS and let P = 0. (0) be denoted simply by P, we hae = i. () With equation takes the familia fom F i (2) =. (3) ow suppose ou subset is losing mass, e.g., the ocket poblem. P = M, so P changes duing each time inteal as the system loses mass dm and/o ealizes a new. Theefoe, = dm + Md. (4) As is appaent fom the deiation of equation 3, its RHS changes in this instant also, since the uniesal complement to ou system has acquied a new subset of mass dm with, say, elocity u. Theefoe, equation 3 becomes (pe ou deiation theeof) dm d dm + M = u + (5) when the system unde inestigation is losing mass. [This equation can be used to account fo a system gaining mass by assuming dm is positie and dm is negatie.] Again, dm/ = dm/ so we hae dm d dm + M = u +. (6) Contay to H&R s statement, we did deie a geneal expession fo ewton s second d( M) law fo aiable mass systems by teating the mass in = = as a aiable.

4 One must ealize the oigin of the expession (equation 3) to undestand how to use it in such cases. Fo claity, it s pobably wise to intoduce ewton s second law in the fom Uni = 0. (7) As an aside, we show that abitaily changing the definition of ou subset in a timedependent fashion does not change ou dynamics pe equation 6. Conside the case whee M is a function of time as in the ocket poblem, but the fuel is not actually ejected. That is, duing each instant we ceate a new subset of the uniese by atificially edefining ou system. Equation 6 becomes since u. This gies O, without enal foces dm d dm + M + (8) d M =. (9) d M = 0 as must be the case, since thee is no physical basis fo the time dependency of M. QED Finally, we see that equation 6 yields (20) d dm dm M = ( u ) + F e + F (2) whee I used u e, whee u is the exhaust elocity elatie to an enal fame, is the ocket s elocity elatie to that same fame and e is the elocity of the exhaust elatie to the ocket. That follows fom: fg RG whee fg = u is the elocity of the fuel elatie to some enal fame, e.g., the gound, RG is the elocity of the ocket elatie to the same enal fame and fr e is the elocity of the fuel elatie to the ocket. ote the ode of the subscipts is impotant and in geneal we hae: AB + fg BC + GR AC fr

5 i.e., the elocity of A elatie to B plus the elocity of B elatie to C is equal to the elocity of A elatie to C. I also used the fact that CB = BC, i.e., the elocity of C elatie to B has the same magnitude but opposite diection of the elocity of B elatie to C. The tem F = 0 dm e on the RHS of equation 2 is called the thust and in cases whee dm, since is negatie, we see that e is oiented in the opposite diection of d, as expected. M s acceleation,