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) = [(v0cosθ)t]ˆı + [(v0sinθ)t − 1 2 gt2 ]ˆ (1) v(t) = [v0cosθ]ˆı + [v0sinθ − gt]ˆ (2) Here, v0 is the initial velocity, θ 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) = [ v0cosθ γ (1 − e −γt)]ˆı + [( v0sinθ γ + g γ 2 (1 − e −γt) − g γ t]ˆ (3) v(t) = [v0cosθe−γt]ˆı + [v0sinθe−γt − g γ (1 − e −γt)]ˆ (4) Here, γ is the linear drag coefficient. The linear drag coefficient, γ, is calculated from the following equation. γ = 6β πρD2 (5) Here, β is the ballistic coefficient, ρ 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	θ	48◦
Baseball Mass	m	5.125oz
Diameter of a baseball	D	2.86in
Ballistic coefficient	β	0.2 Ns/m^2

Answer 1

Parameter	Variable	Values	Value SI
Initial Velocity	v0	105.8 mi/hr	47.296832 m/s

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

Parameter	Variable	Values	Value SI
Initial Launch Angle	θ	48◦	0.8377 rad

SI Unit for angle is rad. 180 degrees = pi radians  

-> θ = 48 degrees = (48*pi/180) radians = 0.8377 rad OR 48°*π/180° = 4π/15 radians

Parameter	Variable	Values	Value SI
Baseball mass	m	5.125oz	0.1453 kg

SI Unit for mass is kilogram. 1 oz = 0.0283495 kg

-> m = 5.125oz = (5.125 * 0.0283495) kg= 0.1453 kg

Parameter	Variable	Values	Value SI
Diameter of a baseball	D	2.86 in	0.0726 m

SI Unit for length is meter. 1 inch = 0.0254 m

-> m = 2.86 in = (2.86 * 0.0254 ) m= 0.0726 m

Parameter	Variable	Values	Value SI
Diameter of a baseball	β	0.2 Ns/m^2	0.2 Ns/m^2

Everything here is in SI unit. N is equivalent to kg⋅m⋅s⁻²