In this assignment you will study the effects of air resistance in 2-D projectil
ID: 3307703 • Letter: I
Question
In this assignment you will study the effects of air resistance in 2-D projectile motion. Note: Your values for numbers should always have units where appropriate and you need to show all steps in your calculations. You will: a) Evaluate position and velocity vectors for 2-D motion with and without air resistance. b) Derive the time of flight of a particle without air resistance. c) Derive the maximum range of a particle without air resistance. d) Compare the maximum ranges of a particle with and without air resistance. e) Quantify the effect that air resistance has on a particles maximum range. f ) Summarize your results. As we have seen before the time dependent position and velocity vectors for 2-D projectile motion can be given by Equations 1 (position vector) and 2 (velocity vector): r(t) = [(v0cos)t]ˆ + [(v0sin)t 1 2 gt2 ]ˆ (1) v(t) = [v0cos]ˆ + [v0sin gt]ˆ (2) Here, v0 is the initial velocity, is the launch angle, and g is the acceleration due to gravity. The derivation of the same equations for particles with air resistance involves integration of a much more complex acceleration vector, they are given by Equations 3 (position vector) and 4 (velocity vector): r(t) = [ v0cos (1 e t)]ˆ + [( v0sin + g 2 (1 e t) g t]ˆ (3) v(t) = [v0coset]ˆ + [v0sinet g (1 e t)]ˆ (4) Here, is the linear drag coefficient. The linear drag coefficient, , is calculated from the following equation. = 6 D2 (5) Here, is the ballistic coefficient, is the density of the baseball, and D is the diameter of the baseball.
In the 2017 World Series, Carlos Correa of the Houston Astros hit a home run with a season record for the highest launch angle. Below are the parameters of his record hit and the values you will use for this problem. In the fourth column, fill in the SI units for the parameters where appropriate (hint: only three parameters need to be converted.).
Part A: Show your work for your unit conversions of the parameters in the space below.
Parameter Variable Values Value SI Initial Velocity v0 105.8 mi/hr Initial Launch ANgle 48 Baseball Mass m 5.125oz Diameter of a baseball D 2.86in Ballistic coefficient 0.2 Ns/m^2Explanation / Answer
SI Unit for Length is meters and for time is seconds. Hence for velocity it will be meter/second
-> v0 = 105.8 miles/hours = (105.8 * 1609.34 m)/ (3600 seconds) = 47.296832 m/s
SI Unit for angle is rad. 180 degrees = pi radians
-> = 48 degrees = (48*pi/180) radians = 0.8377 rad OR 48°*/180° = 4/15 radians
SI Unit for mass is kilogram. 1 oz = 0.0283495 kg
-> m = 5.125oz = (5.125 * 0.0283495) kg= 0.1453 kg
SI Unit for length is meter. 1 inch = 0.0254 m
-> m = 2.86 in = (2.86 * 0.0254 ) m= 0.0726 m
Everything here is in SI unit. N is equivalent to kgms2
Parameter Variable Values Value SI Initial Velocity v0 105.8 mi/hr 47.296832 m/s