Projectile Motion Using Runge-Kutta Methods

Transcription

1 Projectile Motion Using Runge-Kutta Methods Kamalu J. Beamer April 2, Introduction An object s trajectory through a medium is an important phenomenon that has many applications in physics. These trajectories are defined by differential equations. There are many ways to evolve dynamical systems through the evolution of their differential equations. Using the 4th order Runge-Kutta method, a program was developed to calculate the trajectory of a baseball s precession through the air. The simulations were done with different constraint conditions. Data sets were produced for one drag-free environment, two environments with different altitudes and drag forces, and two environments with different altitudes, drag forces and with either head or tail wind. It was determined that a 45 0 home run launch angle was optimal in the vacuum (drag-free) while a 40 0 home run launch angle was optimal in all environments with drag force. It was also determined that at higher altitudes, drag forces are exponentially reduced. The applied drag forces are described in the Theory section and the two environments with different altitudes were NY (altitude = 0 m) and Denver (altitude = 1600 m). After the program was designed to accomodate for drag forces as functions of altitude, additional head and tail winds were applied. Simulations were done in both environments, each with a head and then a tail wind of 15 mph = m s. 2 Theory A baseball s flight through the air is only an interesting case if drag forces are applicable. A baseball s flight through a vacuum follows simple parabolic motion, as is seen in Figure 3; the red line represents the no drag case. Figure 1 [4] is a free-body diagram of a baseball in flight; velocity is not a force, the arrow is to indicate the direction of the velocity. In the drag free case, gravity is the only force affecting the ball, therefore the velocity is constant throughout the entire flight of the ball. With only gravitational force, the baseball follows simple parabolic motion. These phenomena are resultant of Newton s Laws [4], most importantly the Law 1 an object continues in its initial state of rest or motion with uniform velocity unless it is acted on by an unbalanced, or net external force. [4]. 1

2 Figure 1: A baseball s precession through the air with an initial velocity, and two applied forces. With the introduction of drag force, the dynamical system acts significantly different. The effect of air resistance is important for... a high flying baseball hit deep to the outfield [2]; the air resistance or drag force is opposite the projectiles velocity. The Prandtl expression for the motion of a particle in a medium in which there is a resisting force proportional to the speed or two the square of the speed [2] is as follows. F drag = 1 2 C wav 2 ρ In the previous equation C w is drag coefficient, A is the cross-sectional area of the object, ρ is the air density, v is the velocity of the object. All values relative to the lab are given in Table 1 [1]. The final drag force is an additional force applied to the baseball during it s flight. One further modification must be made to the previous equation before we can apply it to the baseball. We are also interested in observing a baseball s trajectory at different altitudes, yet there is no height dependence in the previous equation. To remedy, we equate ρ = ρ 0 exp h/h0 [1] with h as the altitude and h 0 (scale height) as a constant found in Table 1. Baseball Mass kg Baseball Area m 2 Drag Coeff (C w ) 0.35 Air Density (ρ 0 ) 1.22 kg Typical HR Velocity 49.0 m s Scale Height (h 0 ) 7000 m Table 1 - Input values of trajectory-calculating program. With the revised drag force, all applicable forces were cast on the baseball in flight. The following equation is the final form of Newton s force law applied to the baseball. F total = F grav + F drag Since both forces are in the opposite direction of the baseball s initial velocity, they are both considered negative. With this characterization, we obtain the following. 2

3 m dv dt = -mg C wav 2 ρ 0 exp h/h0, dx dt Evidently, these are the equations of motion converted to the form utilizable by RK4 [1]. The 4th order Runge-Kutta method is a more stable version of the 2nd order Runge-Kutta method which was not applied in this lab. General Runge-Kutta methods are a form of numerical analysis that utilize an iterative method for differential equation solution approximation. Proper Runge-Kutta implementation and function definition can save significant amounts of code, and was therefore imbedded into the developed program. Other (non-rk4) functions were also defined and can be seen on my course webpage (www2.hawaii.edu/ kjbeamer/) under Lab 9 code. The RK4 function was used in an iterative process in the code wherein new values were calculated. At each consecutive step, the old value was set equal to the newly calculated value so that the systems dynamically evolved, step by step. The following is the 4th order Runge-Kutta (RK4) formulation. k 1 = dt f(t,y) k 2 = dt f(t+ dt 2,y(t)+k 1/2) k 3 = dt f(t+ dt 2,y(t)+k 2/2) k 4 = dt f(t+dt,y(t)+k 3 ) = v y(t+dt) = y(t) (k 1 + 2k 2 + 2k 3 + k 4 ) With this RK4 formulation in the form of a function, the correct gravitational force and drag force were inserted into the functions. The program was used to calculate the trajectory of a baseball in many different atmospheric situations. The only problem is that no error was calculated by the program as I am unsure how to incorporate error into a seemingly perfectly deterministic iterative process. 3 Data/Calculations The projectile code was converted into an executionable and the range of a hit vs. initial angle was calculated (data in Table 2) and graphed in Figure 2. As the drag force is a function of ρ and ρ is further a function of exponential reliance on height, we expect the zero drag case to produce the largest horzitontal range, the Denver case to produce the second largest horizontal range (because Denver sits at a higher altitude), and the NY case to produce the smallest horizontal range. The data found in Table 2 confirms these predictions. 3

4 θ 0 (deg) No Drag Dist(m) Drag in NY Dist(m) Drag in Denver Dist(m) Table 2 - Simulated values of distance travelled as a function of initial launch angle for the no drag case and the two drag cases at different altitudes. Evident in Table 2 are the maximum home run distances of different atmospheric conditions; one with no atmosphere, one with NY s atmosphere and one with Denver s atmosphere. Because air density is exponentially related to altitude, the maximum home run distances will also vary as a function of altitude. This explains why the maximum distance without drag is 2 times larger than the maximum distance with drag in NY and 1.8 times larger than the maximum distance with drag in Denver. The following graph represents the total horizontal range spanned by baseballs launched at different initial angles. As theoretically expected, the red line (no drag case) follows a perfect parabolic trajectory. Also as expected, a home run hit in Yankee Stadium experiences more drag force and therefore covers less horizontal range than a home run hit at Coors Field. It is interesting to note that even though the two environments in which the simulations take place differ by 1600 m, the range of values of horizontal distance covered by the baseballs are always within 11 meters of each other. 4

5 Figure 2: Trajectories of baseballs launched with the same total speed, but at different initial angles. The red, green and blue lines represent trajectories of increasing drag force. From Figure 2, it is evident that drag significantly affects the range. In the no drag case, optimal horizontal distance is associated with an initial launch angle of 45 0, as is theoretically expected. Interestingly, when drag forces are introduced (in NY and Denver), optimal horizontal distance is associated with an initial launch angle of This has to do with the time that the baseballs are in the air; the longer the duration, the longer the drag force is applied. This explains why an angle slightly smaller than the theoretical expectation produces a traversed distance. A baseball launched at a smaller angle will be in the air for a shorter period, and will be subject to the drag force for a shorter period. The following figure properly displays the affect that drag forces have on trajectory. The seven trajectories with the largest x ranges are the baseballs hit in the ideal, drag free case. The seven trajectories with the smallest x ranges are of baseballs hit with the same initial conditions as the other seven, except the addition of a drag force at sea level. 5

6 Figure 3: The trajectories of baseballs hit in NY, all with the same initial velocity. The addition of a drag force can half the horizontal range of the baseball s path. Once the projectile programs was able to calculate trajectories while including height dependent drag forces, it was further modified to take head and tail winds into account. Table 3 contains the data of the simulations at both altitudes, both with a 15 mph tail wind (wind in the same direction as the ball) and a 15 mph head wind (wind in the opposite direction as the ball). This was a simple adjustment to the code; being that 15 mph is equivalent to m s, this value was simply added to the initial velocity in the tail wind case, and subtracted from the initial velocity in the head wind case. Simple vector addition allows us to make this subtle mathematical change. θ 0 (deg) Denver-tail(m) NY-tail(m) Denver-head(m) NY-head(m) Table 3 - Simulated values of distanced traveled for different launch angles. Data was taken at two altitudes with either a 15 mph tail or head wind. 6

7 The data in Table 3 is what was theoretically expected and is plotted in the following figure. The difference between the red and green trajectories is the altitude difference between NY and Denver. Being that Denver has a lower atmospheric density, baseballs hit in Denver travel greater distances than those in NY. The difference between the red and blue trajectories is the direction of the wind. Being that a tail wind will add to the horizontal component of the baseball s velocity, baseballs in this scenario will travel further distances, as is evident in Figure 4. Figure 4: A plot of the data in Table 3. For each respective tail or head wind case, the baseball launched at the higher altitude location covered a greater range. 4 Conclusion The 4th order Runge-Kutta method again proves to be a valuable numerical analysis technique to be utilized in code. The RK4 s versatility through the insertion of force laws makes it widely applicable to many dynamical systems that are described by coupled differential equations. The projectile program calculated drag trajectories that had smaller ranges than their no drag counter parts. As the drag force depends on air density, experimental concepts agree with theoretical concepts. It was also determined that a 45 0 home run launch angle was optimal in vacuum, while a 40 0 home run launch angle was optimal in all environments with drag. As theoretically expected, it was also determined that at higher altitudes, drag forces are exponentially reduced. 7

8 5 References 1. Gorham, Peter. Physics305: Computational Physics. Index. N.p., n.d. Web. 02 Apr < gorham/p305/p305index.html>. 2. Marion, Jerry B., and Stephen T. Thornton. Classical Dynamics of Particles and Systems. New York: Harcourt Brace and Company, Print. 3. Rainville, Earl D., and Phillip Edward Bedient. Elementary Differential Equations. New York: Macmillan, Print. 4. The Path and Range of a Baseball. ThinkQuest. Oracle Foundation, n.d. Web. 02 Apr < 8