IJAMES Vol. 9, No. 1, January-June 015, Pp. 1-31 Serials Publications New Delhi (India) THE MAXIMUM VELOCITY OF A FALLING BODY M. Rahman and D. Bhatta ABSTRACT: This paper is primarily concerned with a mathematical model using the concept of Newton s law of motion to predict the maximum velocity of a falling body from some altitude above the surface of the earth. This is a real life problem which draws the attention to the scientists and mathematicians for its investigation. This problem was triggered by the great fall of an English Sydiver, Felix Baumgartner. A few articles on falling humans have appeared in some journals including Mathematics Today of the Institute of Mathematics and its Applications, the publisher is none other than the Oxford University Press. The present mathematical model is considered with the variation of the air resistance coefficient (x) as an exponential function of the distance fallen x. The authors have made some interesting analysis with extensive numerical manifestations. In the present paper, we have clearly and elegantly demonstrated a mathematical model initiated by Mahony [5]. We have obtained a beautiful and sophisticated analytical solution which involves the exponential integrals in a complicated manner. We have used here the Maple-18 software including fsolve to compute the various complicated functions contained in the analytic solution. Using the computer software we are able to show two interesting figures with the corresponding numerical results in the form of tables. We have also demonstrated a very simplified mathematical model with constant air resistance coefficient and presented its results in a tabular form using the neat, clear and closed form analytic solution. Keywords: Newton s law of motion, maximum velocity, falling body, distance fallen, altitude, air resistance, Felix Baumgartner, exponential integral, universal gravitation. 1. INTRODUCTION The problem on falling body from some altitude (as is seen fall of a sydiver from the sy) is a very interesting one to investigate when the body experiences air resistance to overcome the gravitational force that pulls the body downward. A few articles on Falling Humans in Mathematics Today [1,, 3, 4] made very interesting analysis using mathematical models. Recently, Mahony [5] studied the problem with some mathematical sophistication with regard to the term air resistance. This problem was triggered by the great fall of the English sydiver Felix Baumgartner some few years ago. In one of the papers an exponential decay with height model was employed to represent, in effect, the motion resistance factor has been examined further to show how it relates maximum velocity to the distance fallen and altitude. Specifically, the motion resistance factor can be represented by a simple two exponentials model that permits a closed form analytical solution to the equation of motion. This solution can be exploited to determine the values for the two parameters that are required to support nown facts. In this fashion, it is possible to replicate some of the results/curves in the published literature. This method, simple as it is, is instructive and shows how a mathematical artifact n can be employed to determine the motion of a falling body that reaches a maximum velocity at some intermediate flight point, after which it slows down [5]. In the present paper, we have developed a mathematical model with clear and elegance of mathematical description. To obtain the solution we need to depend on the computer algorithm to compute the closed form analytical solution that contains considerable sophisticated mathematical functions in integral forms. The closed form solution contains the exponential integral, and we here used the Maple-18 with software fsolve to compute the results. These results are shown in Figure 1 and Figure with the two corresponding Table 1 and Table. This study also contains a small section which deals with when the air resistance coefficient is a pure constant.
M. Rahman & D. Bhatia Table 1 Computational results for distance fallen x(t), velocity v(t), and altitude at various times are presented. This Table shows at what time and altitude, the maximum velocity of the falling body occurs Time Distance fallen Velocity Alititude 0.00 0.00 0.00 40000.00 7.67 114.86 149.04 38857.14 13.11 85.71 09.85 37714.9 17.59 348.57 55.48 36571.43 1.49 4571.43 9.6 3548.57 5.0 5714.9 33.63 3485.71 8.30 6857.14 349.44 3314.86 31.38 8000.00 370.9 3000.00 34.34 914.86 386.01 30857.14 37.3 1085.71 396.14 9714.9 40.08 1148.57 399.99 8571.43 4.96 1571.43 396.76 748.57 45.93 13714.9 385.69 685.71 49.05 14857.14 366.8 514.86 5.4 16000.00 338.59 4000.00 56.19 1714.86 303.64 857.14 60.5 1885.71 63.67 1714.9 65.67 1948.57.19 0571.43 71.90 0571.43 183.40 1948.57 79.48 1714.9 150.78 1885.71 88.58 857.14 15.58 1714.86 99.9 4000.00 106.63 16000.00. EQUATIONS OF MOTION This is a practical problem which states that a body of mass m is dropped from rest in a medium which offers resistance proportional to v where v is the instantaneous velocity of the body. We assume that the gravitational force mg is constant. We need to find the position (distance fallen) and the velocity of the body at any time. In this situation it is convenient to consider the x axis positive downward with the origin at the initial position of the body. The weight mg of the body then acts in the downward (positive) direction, but the air resistance m(x)v, where (x) is a positive function of the distance fallen x in which (x) = exp( + ), acts to impede the motion. Here and are two unnown constants to be determined in course of calculation. When v > 0 the air resistance is in the upward (negative) direction, and is thus given by m(x)v. Thus using the Newton s law of motion, the equation of motion governing a body falling under the influence of gravitational force alone and subjected to air resistance that varies with the square of its speed may be written in the form dv( x) mv( x) mg m( x) v ( x) (1) where dt v( x) Note that
The Maximum Velocity of a Falling Body 3 Table Computational results for distance fallen x(t), velocity v(t), and altitude at various times are presented. This Table shows at what time and altitude, the maximum velocity of the falling body occurs Time Distance fallen Velocity Alititude 0.0 0.0 0.0 46000.00 7.67 1150.00 149.9 44850.00 13.10 300.00 11.89 43700.00 17.53 3450.00 59.7 4550.00 1.38 4600.00 98.95 41400.00 4.83 5750.00 333.51 4050.00 7.99 6900.00 364.17 39100.00 30.9 8050.00 391.47 37950.00 33.69 900.00 415.55 36800.00 36.33 10350.00 436.1 35650.00 38.87 11500.00 45.88 34500.00 41.34 1650.00 464.58 33350.00 43.79 13800.00 469.87 300.00 46.5 14950.00 466.81 31050.00 48.79 16100.00 453.01 9900.00 51.49 1750.00 46.03 8750.00 54.49 18400.00 384.14 7600.00 57.99 19550.00 37.85 6450.00 6.39 0700.00 61.77 5300.00 68.6 1850.00 195.63 4150.00 76.39 3000.00 141.57 3000.00 87.6 4150.00 105.81 1850.00 100.99 5300.00 83.71 0700.00 The initial conditions are given by: d x dv dv dv v. dt dt dt At t = 0, x(0) = 0, v(0) = 0. (3) Equation () is presented in a format that enable one to see how the time factor may be determined by numerical integration once the velocity is nown as a function of distance. In the above, m denotes the point mass of the body, g(= 9.78)m/s ) denotes the acceleration due to gravity, v(x) denotes its downward velocity at a distance x from its start point and (x) denotes a motion resistance factor that varies with distance fallen; it may be related to the drag coefficient from information provided in [1]. However, this relationship will not be explored in detail in this paper. Rather, motion resistance term will be assumed heuristically in the exponential form () (x) = exp( + x) = (0) exp(x); (0) = exp( ) (4) and it will be shown how values of the parameters and may be chosen to ensure that maximum velocity and distance details, assumed nown, can be met. As a matter of convenience, equation (1) will be rewritten in the form
4 M. Rahman & D. Bhatia Figure 1: Graphical representation of the velocity and the altitude of the falling body with respect to time corresponding to the numerical values in Table 1 d ( v ) ( x )( v ) g The integrating factor of equation (5) is exp exponential integral form is given by (5) ( x). Now using the initial conditions (3), its solution in g (0) (0) v Ei Ei exp( ( x)/ ) (6)
The Maximum Velocity of a Falling Body 5 y y z e where Ei ( z) dy y and e Ei ( z) dy. [see Abramowitz and Stegun [6]]. The detailed calculation z y of the solution of equation (5) is given in Appendix A. In the above, E i denotes the usual exponential integral [6] and its series expansion is z Ei ( z) ln( z) ( n) n! (7) Here, z is real and positive and = 0.577156649 is Euler s constant, which is not needed explicitly in what follows. It is of interest to note that when the motion resistance factor (x) 0 and the solution given by equation (6) simplifies to the well-nown result v =gx, where the velocity continues to increase with the distance fallen. However, if the motion resistance factor is not zero it is conceivable that the velocity might reach a limiting value at some specified distance. Although we have adopted (4) for the resistance factor it should be appreciated that other formats might not allow for a local maximum. Then, it is of interest to see how values for the input parameters and can be chosen to ensure agreement of results from the model with nown data. This is addressed below. 3. DETERMINING VALUES FOR AND If the velocity reaches a nown maximum, v m, say, at some nown point, x m, so too does its square, the derivative of which must be zero. It follows from equation (5) that this occurs if Also, after first writing and m n1 n g ( xm ) (8) v gxm ( xm ) xm (9) v m X 1 x (10) m one can return to equation (6) and at the point of greatest speed, rewrite it in a more convenient form exp( X ) Ei ( X ) Ei ( X exp( 1/ X ) X For the specified input conditions of maximum velocity and associated distance, may be treated as a nown quantity in this equation and X (and hence ) is a solution to be determined, if it exists. If it does, it will ensure that the input conditions above can be met. For the motion resistance factor specified in equation (4), it follows that at the point of maximum velocity (0)/(x m ) = exp( 1/X), and so the argument of the last exponential integral term on the right-hand side of the equation (11) can be replaced by X exp( 1/X). Thus the equation (11) can be expressed as (11) exp( X ) Ei ( X ) Ei ( X exp( 1/ X )) X (1)
6 M. Rahman & D. Bhatia Figure : Graphical representation of the velocity and the altitude of the falling body with respect to time corresponding to the numerical values in Table Once is nown from (9), we can determine X using (1) and hence from (10). It is possible to obtain a value for from (4), after taing logarithms and using (8), in the following form g xm ln vm Values for and obtained in this manner can be employedin equation (6) to tabulate the velocity as a function of either distance fallen or altitude, again using the series expansion (7) for the exponential integral. (13)
The Maximum Velocity of a Falling Body 7 4. RESULTS AND DISCUSSION An exercise was carried out to test the results of this model [5]. Here we use Maple-18 for our computational wor. To compute the zeros of (1), we use various function calls from Maple software including fsolve. To compute the velocity, we use equation (6) and the Ei function of Maple. We examined two cases. Example 1 For the first case, it was assumed that the fall started at an altitude of 40,000 m above mean sea level (MSL) and that a maximum velocity of 400 m/s occurred at an altitude of 8,500 m (MSL). We use g = 9.78 m/s. The relation (9), yields the value of = 1.405875. Solution of (1) is obtained as X = 0.389508. Subsequent values of and are given by =.350034 10 4 and = 1.69965. Computational results for distance fallen x(t), velocity v(t), and altitude at various times are presented in Table 1. Example For the second case, it was assumed that the fall started at an altitude of 46,000 m above mean sea level (MSL) and that a maximum velocity of 470 m/s occurred at an altitude of 3,000 m (MSL). We use g = 9.78 m/s. The relation (9), yields the value of = 1.39655. Solution of (1) is obtained as X = 0.56867. Subsequent values of and are given by = 3.16494376 10 4 and = 14.456047. Computational results for distance fallen x(t), velocity v(t), and altitude at various times are presented in Table. 5. CONCLUSION The interested reader might also consider using these models to see to what extent they can support the revised data from Felix Baumgartner s fall. With respect to his fall, it should be appreciated that a falling sydiver can control his/her drag coefficient (and hence motion resistance factor) simply by changing the cross-sectional area that he/she presents to the direction of motion. For example, if it is a cruciform shape the area is as large as possible whereas if it is a head-first dive it is perhaps as small as possible. The one aspect might be several times larger than the other. Thus, for a given atmosphere, the drag coefficient and hence motion resistance factor might vary almost instantaneously by a significant amount, perhaps at the whim of the sydiver, intended or otherwise. These factors have not been included in the model but that is not to say that a two-stage flight model of the type proposed here cannot be employed to produce a more accurate model. To do this, more detailed information concerning the great fall would be required. APPENDIX A Solution (6) for variable air resistance coefficient (x)) The integrating factor (If) of equation (5) is obtained as follows: If e e Multiplying equation (5) throughout we obtain x ( ( x) (0) e ( x) e. d [ v e ] ge ( x)/ ( x)/ Now integrating this differential equation in the range < u x treating u as the dummy variable we obtain
8 M. Rahman & D. Bhatia ( ( x)/ x ( ( u)/ ( ) v x e g e du C The value of C is determined using the initial condition (3), i.e. the value at x(0) = 0 which yields 0 ( ( u)/ C g e du And hence we get g x ( ( u) / 0 ( ( u) / v ( x) [ e du e du] ( ( x) / e With the substitution (u)/ = y such that equation in the form dy du y and after a little reduction we can rewrite the above y x ( ( u) / 0 ( ( u) / ( ) ( ( x) / g e e v x e du e du e y y y g ( x ) (0) Ei Ei exp( ( x) / APPENDIX B A special case for constant air resistance coefficient The equation of motion governing a body falling under the influence of gravity alone and subjected to air resistance that varies with the square of its speed may be written in the form [7] where ( ) mv( x) dv x mg mv (14) v dt (15) Here the air resistance coefficient is assumed constant. Equation (15) is presented in a format that enable us to see how the time factor may be determined by numerical integration or otherwise once the velocity is nown as a function of distance (altitude). In the above, m denotes the point mass of the body, g(= 9.81)m/s denotes the acceleration due to gravity, v(x) denotes its downward velocity at a distance x from its starting point of descent and denotes a constant motion resistance factor (coefficient) which may be related to the drag coefficient (used in fluid dynamics).
The Maximum Velocity of a Falling Body 9 The initial conditions are: At t 0, x(0) 0, v(0) 0 (16) dt Solution As a matter of convenience, (14) can be rewritten in the following form: d ( v ) ( v ) g (17) The solution of (17) can be written at once as v g Ce (18) x Using the initial condition, we obtain the solution v in terms of x as v g [1 e ] (19) x Now this solution (19) can be written in terms of time variable t, and we illustrate the steps below. In terms of time variable we have 1 exp( x) exp( x) exp( x) 1 g dt g dt (0) We now substitute exp(x) = z such that exp(x) = 1 dz. reduced to after integration, Using this information, equation (0) can be dz gt (1) z 1 The solution is simply ln 1 z z gt () Equation () can be simplified a great deal, and we obtain the following simple form z cosh gt z e cosh gt
30 M. Rahman & D. Bhatia Table 3 Non dimensional analysis of the distance fallen by the body X = x and velocity V v g of the falling body with respect to non dimensional time T g t. T X = Distance Fallen V = Velocity 0.0 0.00000000 0.00000000 1.0 0.43378083 0.76159416.0 1.350075 0.9640758 3.0.3093851 0.99505475 4.0 3.3077883 0.9993930 5.0 4.306898 0.9999090 6.0 5.30685896 0.99998779 7.0 6.30685365 0.99999833 8.0 7.3068593 0.99999977 9.0 8.3068583 0.99999997 10.0 9.306858 0.99999999 11.0 10.306858 0.99999999 1.0 11.306858 0.99999999 13.0 1.306858 1.00000000 14.0 13.306858 1.00000000 15.0 14.306858 1.00000000 1 x ln cosh gt (3) g v dt (4) tanh gt Thus at certain instant, the position (distance fallen) of the falling body is given by (3) and at the same instant, the velocity of the falling body is given by (4). It is worth noting that these physical solutions are dimensionally consistent. Let us define the dimensionless variables as follows: T gt, X = x and V / gv, where T, X and V are, respectively, the dimensionless time, position and velocity of the falling body. Then in dimensionless form the position X and velocity V can be written with respect to the dimensionless time as X = ln(cosh T ) V = tanh T The only unnown parameter in this problem is simply the air resistance coefficient,. The following Table 3 shows that the maximum dimensionless velocity of the falling body attains 1 at the dimensionless time 15. And the maximum distance fallen by the body is 14.306858 at the same time units. The actual dimensional quantities can be obtained once we now the exact value of the coefficient of air
The Maximum Velocity of a Falling Body 31 resistance. We now the value of g = 9.81 m/s. But we do not now the value of. Some rough estimate of this parameter is = 0.00008. With this value we can determine the maximum velocity and maximum distance fallen by the body from its starting position of fall from the sy. This parameter is ubiquitous. It is not a fixed parameter. It varies from problem to problem. At the same time it depends very much on the mathematical model chosen to determine the solution. ACKNOWLEDGEMENTS We are grateful to the Faculty of Computer Science at Dalhousie University for giving access to its computer facilities. Thans are extended to Dr Denis Riordan, Associate Dean of the Faculty of Computer Science for his interest and encouragements. REFERENCES [1] Wheeler, P. (013). Falling Humans, Mathematics Today, Volume 49, Number 4, 176-181. [] Stevens, A. (013). Falling Humans, Mathematics Today, Volume 49, Number, p. 90. [3] Neve, R. (013). Falling Humans, Mathematics Today, Volume 49, Number, p.9. [4] Mahony, J. D. (013). Baumgartner s Fall from Grace, Mathematics Today, Volume 49, Number 3, pp. 141-14. [5] Mahony, J. D. (014). The Maximum Velocity of a Falling Body, Mathematics Today, Volume 50, Number, April, 96-97. [6] Abramowitz, M. and Stegun, I. A. (1970). Handboo of Mathematical Functions, Dover Publications. Inc., New Yor. [7] Boyes, William, E. & DiPrima, Richard C. (1969). Elementary Differential Equations and Boundary Value Problems, Second Edition, JohnWiley & Sons, Inc., New Yor. M. Rahman Faculty of Computer Science, Dalhousie University, Halifax, Canada D. Bhatta Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, USA