Projectile Motion with Air Resistance (Nuerical Modeling, Euler s Method) Theory Euler s ethod is a siple way to approxiate the solution of ordinary differential equations (ode s) nuerically. Specifically, those ode s for which initial conditions are known. Consider the ode which has solution dy dx = f (x) (1) and reference Figure 1. y = f(x) Figure 1: Euler s ethod. The Euler ethod is a nuerical ethod; with known (x o, y o ), a chosen x, and the approxiation Equation 1 becoes dy dx y x 1
y = f (x) x Thus x 1 = x o + x y 1 = y + f (x o ) x The process is then be repeated; the general algorith is x n = x n 1 + x () y n = y n 1 + f (x n 1 ) x (3) Physics is all about rates of change so this ethod can be used to solve a wide variety of physics probles. Reeber that a = dv dt and v = dx dt and consider an object oving in one diension subject to a constant rate of acceleration a; it starts fro rest (x o = v o = t o = 0). Then t 1 = 0 + t x 1 = 0 + 0 t v 1 = 0 + a t a 1 = a and in general t n = t n 1 + t x n = x n 1 + v n 1 t v n = v n 1 + a n 1 t a n = a This is rearkably easy to ipleent in a spreadsheet such as Excel. The first 10 steps of the proble are shown in Figure ; here, the forulas are displayed in the cells rather than the results to better deonstrate the process [you can also check your solution analytically]. Note that the forulas in Row 3 were copied down the rest of the way. This is the beauty of Excel - with judicious use of relative and absolute cell references the first iterative row can be copied down as far as needed - which ay be several thousand steps! Although the ethod will work on any ode it is ost often eployed on those difficult or ipossible to solve analytically. The acceleration of the object need not be constant; it could be a function of any nuber of variables - position, velocity, or tie. As long as initial values can be quantified it can be used. But, a word of warning - all nuerical ethods give but approxiate solutions and are subject to accuulation error. You can iniize this with very sall t.
Figure : Excel ipleentation of the Euler ethod. Air Resistance on a Projectile Assue that air resistance for a projectile is proportional to the square of the projectile velocity; i.e., the drag force on the projectile is given by where F = Dv and v = v x + v y D = ρca Here, ρ is the density of air, C a diensionless constant called the drag coefficient (0. to 1.0 for baseballs, tennis balls, etc.), and A the cross-sectional area of the projectile. At any point in the otion of the projectile, the drag force is directed opposite to the direction of the velocity, as shown in Figure 3. Figure 3: Drag Force on Projectile 3
The net force on the projectile in each diension is Fx = Dv cos θ = a x = Dv(v cos θ) Fy = g Dv sin θ = a y = g Dv(v sin θ) thus the corresponding accelerations are just a x = vv x a y = g vv y Fro an Euler ethod standpoint this would be ipleented as a nx = v n v nx (4) a ny = g v n v ny (5) Obviously the x and y diensions ust be solved separately. If you know the initial values x o, y o, v xo, and v yo then the ethod can be used. Procedure In this experient you will solve the probles below involving projectiles subject to air resistance. You will do this with an Euler ethod worksheet. At the top of the worksheet will be the basic paraeters g, r, C, ρ,, v o, θ (in degrees!), and t. Include the interediate paraeters A and D calculated with these values. Below this start the table of Euler calculations; it will contain 8 coluns - Step, t, x, v x, a x, y, v y, and a y. The first row is initial values for all quantities; subsequent iterations (rows) are deterined with the Euler equations. These equations should reference the cells at the top of the worksheet; this way you can solve all of the probles by just changing the paraeters and/or initial values at the top. Each tie, scan your table to obtain the answer and write it at the top of the table to the right of your constants (include the t used in the proble). Before you eail e your Excel file, delete all but about 5 rows of your Euler table - this will be easier to eail. When I open your file I will recopy your forulas down. 1. As a test progra find the trajectory of a baseball with and without (D = 0) air resistance. The radius of the baseball is 0.0366, and A = πr. Its ass is 0.145kg, the drag coefficient is 0.5 and the density of air is 1.kg/ 3. The baseball is given an initial velocity of 50/s at an angle of 35 above the +x-axis. With drag the range of the baseball is just over 100; without drag the range is alost 40.. The hoe field for the Colorado Rockies is in Denver (altitude 1 ile), where the air density is only 1.0kg/ 3. How uch farther will the baseball in the trial progra go copared to sea level? 4
3. Estiate the axiu distance a huan being can throw a ping-pong ball. A ping-pong ball has a radius of 0.019 and a ass of 0.004kg. Use a drag coefficient of 0.5 and a density of 1.kg/ 3. The fastest pitchers can throw at about 100i/h. Why can a baseball be thrown uch farther than a ping-pong ball? 4. In tennis, 100i/h is a very good serving speed. How fast is the ball oving when it crosses the baseline in the opposite court, about 4 distant? Use a ass of 0.055kg, a radius of 0.031, a drag coefficient of 0.75, and a density of 1.kg/ 3. The ball leaves the racquet oving horizontally and does not hit the ground until after it crosses the opposite court baseline. 5. Decide whether or not air resistance really is negligible in a typical Projectile Motion lab experient. The projectile sphere has a ass of 16g, a diaeter of 1.5c, a sall C (i.e., closer to 0. than 1), and an initial velocity of 5.00/s. For a launch angle of 45, exaine the otion in both cases drag and no drag. What is the ratio Is air resistance negligible or not? range drag? range 5