Altitude measurement for model rocketry

Transcription

1 Altitude measurement for model rocketry David A. Cauhey Sibley School of Mechanical Aerospace Enineerin, Cornell University, Ithaca, New York I. INTRODUCTION In his book, Rocket Boys, 1 Homer Hickam describes how he his hih-school friends used measurements of the time-of-fliht of their homemade rockets to infer the maximum altitudes attained by these vehicles. They related the time of fliht to the maximum altitude usin the simple ballistic timin formula that nelects the effects of dra forces. This simple formula can ive surprisinly accurate results. The reason for the accuracy of this simple formula is investiated here usin an analysis that takes into account the dra of the vehicle. The analysis is presented first for the case for which the vehicle dra coefficient air density are both constant. In this case, the effects of dra are characterized by a sinle parameter with the dimensions of velocity, which can be identified as the terminal velocity for free fall of the vehicle. The equations for a vertical, impulsive launch are interated to ive a closed form solution for the fliht time as a function of maximum altitude. This result can be used to express the fractional error in the fliht time predicted by the simple ballistic formula for a iven maximum altitude as a function of a sinle dimensionless Froude number, defined as the ratio of the terminal velocity in free fall to the square root of the product of the ravitational acceleration the maximum altitude. An asymptotic expansion is developed for lare values of the Froude parameter is found to be accurate even for Froude numbers of order unity. This result implies that the fractional error in the ballistic timin formula is small even for Froude numbers of order unity. A simple physical arument is iven for the accuracy of the ballistic timin formula. This arument holds qualitatively when the terminal velocity of the vehicle is not constant, for example, when the vehicle dra coefficient is a function of Mach number or the air density varies with altitude. The analysis is repeated for the case in which the air density varies exponentially with altitude, numerical results confirmin the accuracy of the simple ballistic timin formula when the terminal velocity is not constant are also presented. II. CONSTANT TERMINAL VELOCITY The maximum altitude achieved by a model rocket can often be estimated from the time of fliht. We here assume that the burn time of the rocket motor is much less than the total fliht time, that is, the rocket is impulsively iven an initial vertical velocity v 0. In this case the only forces actin on the rocket are ravity aerodynamic dra. Because rockets are hihly streamlined, it is reasonable, as a first approximation, to nelect the dra. In this case the rocket is decelerated at a constant rate until it reaches its maximum altitude, then is accelerated at the same constant rate until it strikes the round. If T is the total fliht time, the maximum altitude h max is determined in this approximation by h max = 1 T/ = T 8, 1 where is the assumed constant acceleration of ravity. 3 If we denote the total fliht time based on Eq. 1 as T 0,we have T 0 = 8h max. We refer to this result as the ballistic timin formula. When the effect of aerodynamic dra is included, the ascent descent phases of the fliht must be analyzed separately. Durin the ascent phase the equation of motion describin the evolution of the manitude of the velocity v is = D/m, dt 3 where D is the aerodynamic dra force m is the mass of the vehicle. The dra force D can be characterized in terms of the dra coefficient C D as 4 D = C D S 1 v, where S is the effective area for the vehicle is the density of the air. The equation of motion can then be written as = dt. 1+ C D S/m v 5 Durin the descent phase of the vertical fliht the ravitational force acts in the direction of motion or dt = D/m, 1 C D S/m v = dt American Association of Physics Teachers 47

2 Equations 5 7 show that the only way in which the parameters of the vehicle enter the equations of motion is in the combination m C D S, which has a simple physical interpretation. A body in free fall in a uniform atmosphere asymptotically approaches its terminal velocity. If we equate the weiht m of the vehicle its dra D, we obtain the terminal velocity v term as v term = m C D S. In terms of this terminal velocity Eqs. 5 7 can be written as = dt 1+ v/v term 10 1 v/v term = dt If we assume that the dra coefficient the air density are constant, then the terminal velocity also is constant Eqs can be interated to ive the ascent time t 1 descent time t as t 1 = v term arctan v 0/v term 1 t = v term ln 1+v f/v term 1 v f /v term, 13 respectively, where v 0 v f are the initial final velocities. For the ballistic trajectory v f =v 0. The initial final velocities can be related to the maximum altitude by usin the fact that v =± dh dt, 14 where h is the altitude, which recasts Eqs into the form v = dh 1+ v/v term 15a v = dh. 1 v/v term 15b For constant terminal velocity Eq. 15 can be interated in closed form to ive v 0 = v term e h max /v term 1 16 v f = v term 1 e h max /v term. 17 Equations 1 16 can be combined to ive t 1 = v term arctan e h max /v term 1, 18 Eqs can be combined to ive t = v term ln 1+ 1 e hmax/vterm e h max /v term In the limit v term Eqs reduce to the ballistic results t 1 = t = h max. 0 Equations ive the ascent descent times, respectively, for fliht to a iven maximum altitude h max for a iven terminal velocity. The total fliht time T is the sum of the ascent descent times T = t 1 + t = v term arctan e h max /v term ln 1+ 1 e h max /v term 1 1 e h max /v term. 1 The results of Eqs. 18, 19, 1 are compared in Fi. 1 with the predictions of the simple, zero-dra estimates iven by Eq. assumin that = m/s v term =55 m/s. The value of v term is probably representative of the more ambitious of Mr. Hickam s steel-cased model rockets. 1 Fiure 1 shows that the simple ballistic formula of Eq. ives remarkably accurate estimates for the maximum altitude when the total fliht time is used. 5 This accuracy results from the effective cancellation of the errors introduced by the effect of dra in the ascent descent phases of the fliht. This cancellation can be understood by notin that the dra force can be interpreted as increasin the effective acceleration of ravity durin ascent, decreasin it durin descent. If the ascent time alone were used in conjunction with the ballistic timin formula to estimate the altitude, then the maximum altitude would be underpredicted. Conversely, if the descent time alone were used, the maximum altitude would be overpredicted. These errors can be fairly lare, even for terminal velocities as hih as the assumed value of v term =55 m/s. For this value, the error in the estimated altitude based on the ascent time can be on the order of 10% for altitudes of about 000 m, correspondin to ascent times of about 0 s. In contrast, when the total fliht time is used, the cancellation is so effective that the error in the estimated altitude is less than % for altitudes up to 8000 m, correspondin to total fliht times of 80 s. Our analysis is based on several assumptions, namely that the burn time of the rocket motor can be nelected, that both the air density the dra coefficient are constant. In reality, the air density may decrease by about 0% at an altitude of 000 m, by more than 60% at 8000 m. Also, compressibility effects are likely to cause sinificant chanes in the dra coefficient for the speeds considered here. 4 A more accurate analysis could take these factors into account for specific cases, but the tendency for cancellation of errors we have noted remains as lon as the total fliht time is used. 473 Am. J. Phys., David A. Cauhey 473

3 Fi. 1. Relation between the maximum altitude the fliht time for a rocket with a terminal velocity of v term =55 m/s. III. ACCURACY OF THE IDEAL ESTIMATE A comparison of Eqs. 1 shows that the fractional error involved in usin the ballistic timin formula is a function only of the dimensionless parameter F = v term hmax. The symbol F is used in Eq. because of its similarity to the Froude number, which appears in the analysis of fluid flow problems in which ravity is important. 6 In terms of F, we write T T 0 = F T 0 arctan e /F lo 1+ 1 e /F e /F 3 This ratio is plotted as the solid line in Fi.. An accurate asymptotic approximation to Eq. 3 can be developed in the limit of lare F. We have arctan e /F 1 F 1 6 F F 5 + O F 7, lo 1+ 1 e /F 1 F e /F 3 F F 5 4a + O F 7, 4b so the asymptotic representation of Eq. 3 for F 1 is T T 0 1 T 0 10 F 4 + O F 6. 5 This result is plotted as the broken line in Fi.. If the fractional error in the ballistic timin formula is evaluated for the up-le or down-le of the fliht alone, we find T T0 1 T 0 up/down 6 F F 4 + O F 6. 6 Thus, the leadin contribution of dra to the elapsed time for either the up-le or down-le of the fliht is proportional to F. These leadin terms exactly cancel when the total fliht time is evaluated, leavin an error proportional to F 4. Note that the timin error is less than 1% for F 1. Fiure can be used to determine the maximum altitude for which the ideal, ballistic formula predicts the fliht time to a iven accuracy. For example, if we wish to have a maximum error of 5% in the fliht time, Fi. shows that the minimum value of F is F=0.60. For the example vehicle with v term =55 m/s, the maximum altitude is h max = v term F = 55 m/s m/s = m For comparison, the asymptotic formula, Eq. 5, ives 1/4 1 F = = 0.639, correspondin to a maximum altitude of h max = v term F = 55 m/s m/s = m Note that althouh the error in the ballistic timin formula is quite small for values of F 1, it beins to increase very rapidly for values of F 1. Also, althouh the asymptotic expansion leadin to Eq. 5 is based on the assumption that F 1, the formula is accurate down to F of order unity. 474 Am. J. Phys., David A. Cauhey 474

4 Fi.. Fractional error in the fliht time as a function of the dimensionless parameter F=v term / hmax. The solid line represents the exact relation iven in Eq. 3 ; the dotted line represents the asymptotic formula iven in Eq. 5. IV. EFFECT OF NONCONSTANT TERMINAL VELOCITY In an actual rocket fliht, the terminal velocity of the vehicle is not a constant because at hih speeds the dra coefficient is a function of the Mach number M=v/a, 4 where a is the local speed of sound, because the air density is a function of altitude. The ballistic timin formula remains accurate even for varyin vehicle dra properties, because the effects of dra on the up-le down-le sements nearly cancel. To see that this cancellation also works when the terminal velocity is not constant, we consider the case of varyin air density. We consider this case, rather than the case of variable dra coefficient C D because a closed-form solution exists for the variable density case for an isothermal atmosphere. If the temperature T 1 of the atmosphere is assumed to be independent of altitude, the variation of density with altitude h follows the law = SL e h/, 30 where SL is the sea-level value of the density the scale heiht =RT 1 /, where R is the as constant. 8 In this case, Eq. 15a for the up-le of the fliht can be recast as dh + e h/ v =, 31 ṽ term where ṽ term = m/ C D S SL is the terminal velocity for sea-level density. If we define the dimensionless variable = e h/, Eq. 31 can be written as 3 d = F v, 33 where F =ṽ term /. Note that this alternative Froude number is based on the scale heiht, rather than the maximum altitude, as in Eq.. The solutions of Eq. 33 can be written as v h = e 1 e h/ /F v 0 + e /F e h/ /F /F e h/ e t t dt, 34 where v 0 is the initial impulsive launch velocity. A similar analysis for the descent from the maximum altitude h max ives /F e h/ v h = e /F e h/ /F e h max / e t t dt. 35 Numerical interation is required to evaluate the exponential interals in Eqs The ascent time t 1 the descent time t for a iven maximum altitude can then be determined by interatin t 1 = v 0 0 t = 0 v f /dt /dt. 36a 36b Typical results are shown in Fi. 3 for an atmosphere havin =8434 m based on stard sea level conditions 9 a 475 Am. J. Phys., David A. Cauhey 475

5 Fi. 3. Fliht times in nonuniform atmosphere with exponentially varyin density. Terminal velocity, based on constant, stard sea-level density, is v term =55 m/s. sea-level terminal velocity ṽ term =55 m/s. As miht be expected, the errors incurred by usin the ballistic timin formula are smaller in this case than for the constant-density atmosphere because the dra force is proportional to the density. The altitude error resultin from use of the ballistic formula is aain seen to be very small for these parameters, as is the fractional error shown as a function of maximum altitude in Fi. 4. We have plotted the fractional error in the altitude h ideal h, 37 h rather than the error in the time iven by Eq. 3. It is interestin to compare the velocity profiles for the constant- variable-density cases. Fiure 5 shows the velocity as a function of altitude for the two cases for fixed values of the sea-level terminal velocity v term =ṽ term =55 m/s an initial impulsive velocity of v 0 =700 m/s. For these conditions, the vehicle reaches a sufficiently hih altitude that the descendin trajectory for the nonuniform density case exceeds the local terminal velocity approaches the sea-level value of the terminal velocity from above, as pointed out in Ref. 10. The authors of Ref. 10 did not mention that for such a trajectory in an atmosphere of monotonically decreasin density, the maximum velocity in descent must equal the local terminal velocity because at the Fi. 4. Fractional error in the altitude for uniform nonuniform atmospheres. The terminal velocity is the same as in Fi Am. J. Phys., David A. Cauhey 476

6 Fi. 5. The velocity profile as a function of the altitude for uniform dotted lines nonuniform solid lines atmospheres. The terminal velocity, for stard sea-level density, is v term =55 m/s the initial velocity is v 0 =700 m/s. The terminal velocity based on the local density is shown by the dash-dotted line. point of maximum velocity the net acceleration must be zero. The terminal velocity based on the local density is also plotted in Fi. 5. Fiure 5 also illustrates how different the velocities are as functions of the altitude on the up down portions of the fliht. Nevertheless, the error in usin the ballistic timin formula to estimate the maximum altitude for this case is only about 1.5% for the exponential atmosphere 1.8% for the constant-density atmosphere. V. CAVEAT An important assumption used in our analysis is that the dra coefficient is the same for the ascent descent portions of the fliht. This assumption can be very danerous for vehicles such as those described by Mr. Hickam because the terminal velocity miht be lare. Mr. Hickam launched his rockets in a remote area of the mountains of West Virinia. The point of this paper has been to explain the accuracy of the ballistic timin formula, not to suest that this technique be used in practical launches, especially in densely populated areas. ACKNOWLEDGMENTS This analysis was motivated by my readin of Homer Hickam s memoir. 1 The book has been adapted for the motion picture October Sky. As pointed out by Hickam, a redeemin merit of the title chane is that the movie title is an anaram for the book title. I am rateful to Mr. Hickam for his book, which brouht back many happy memories of my childhood reminded me of the excitement of the early years of space exploration, which led me in part to study aerospace enineerin. 1 Homer H. Hickam, Rocket Boys Delacorte, New York, Crude estimates suest that the burn times for the most ambitious of Hickam s rockets were the order of 3 to 5 s, compared to total fliht times on the order of 80 s. 3 The assumption of constant ravitational acceleration is quite ood for model rockets. Even for the altitudes reached by Hickam s rockets estimated to be on the order of 5 to 6 miles the ravitational acceleration is reduced by less than one-third of 1%. 4 John J. Bertin, Aerodynamics for Enineers, 4th ed. Prentice Hall, Upper Saddle River, NJ, At least for values of the dimensionless parameter v term / h max that are not too small; see Sec. III. 6 The speed of ravity waves on the free surface of a liquid layer is proportional to, where is the depth of the layer for lon waves or their wavelenth for short waves. Thus, the Froude number v/ plays a role in flows of liquids havin free surfaces that is analoous to the Mach number for the flow of a compressible fluid; see Ref. 7, for example. 7 Frank M. White, Fluid Mechanics, 4th ed. McGraw-Hill, New York, The as constant R can be expressed as the ratio R=k/m, where k is Boltzmann s constant m is the weiht of a molecule of the as. Air is a mixture of several ases whose averae molecular weiht is usually taken to be amu, with a correspondin as constant of R =87.05 m / s K. 9 U. S. Stard Atmosphere U. S. Government Printin Office, Washinton, DC, P. Mohazzabi J. H. Shea, Hih-altitude free fall, Am. J. Phys. 64, Am. J. Phys., David A. Cauhey 477