The frictional drag on a human-sized body moving through the air at (or faster t

Question

The frictional drag on a human-sized body moving through the air at (or
faster than) walking speed is given by Rayleighís velocity-squared law, f=Dv^2:
You can estimate the value of D for a human from the knowledge that a 180 lbs
skydiver falls at a terminal velocity of about 175 mph. (a) Using this value of D;
calculate the force (in lbs) by air friction on a marathon runner traveling at 5
minute per mile pace, a bike racer traveling at 30 mph, and a person subjected
to hurricane strength winds of 120 mph. (b) How many nutritional bars worth of
energy are required to compensate for the work against air friction acting on a
marathon runner going 26 miles at a 5 min/mile pace? (c) Repeat part (b) for
the case where there is a stiff 15 mph breeze blowing on race day, and the first
half marathon is run into a headwind, while for the last half of the race there
is a tailwind. Assume a 5 min/mile pace in both directions.

Explanation / Answer

given frictional drag, F = D(v)^2

given for human

mass m = 180 lbs = 81.64 kg

terminal velocity, v = 175 mph = 78.23 m/s

so at terminal velocity mg = D*v^2

81.64*9.81 = D*78.23^2

D = 0.1308

a. speed of marathon runner, v = 5 min per mile = 12 mph = 5.36 m/s

F = Dv^2 = 3.757 N = 0.8446 lbf ( pound force)

speed of bike rider v = 30 mph = 13.411 m/s

F = Dv^2 = 23.525 N = 5.288 lbf

speed of wind, v = 120 mph = 53.644 m/s

F = Dv^2 = 376.4 N = 84.61 lbf

b. for a marathon runner at a SPEED 5 min / mile for 26 miles = F*d

d = 26 miles = 41842.9 m

F = 3.757 N

energy = 157.203 kJ = 37.572 kCal

energy in one snickers bar , Eo = 250 Cal

so number of snicxker bars required = 37572/250 = 150.288

c. for 15 mph breeze = 6.7056 m/s

for 13 miles

F = D(6.7056 + 5.36)^2 = 19.04169 N

d = 13 miles = 2091.45 m

so E = Fd = 39824.74 J

for the next 15 miles

F = D(6.7056 - 5.36)^2 = 0.2368 N

d = 2091.45 m

E = -F*d = -495.32 J

net energy used = 39329.41849 J = 9399.99 Cal

Number of scnickers needed = E/250 cal = 37.59 snickers

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