The Physics
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Opus in profectus

# Aerodynamic Drag

## Problems

### practice

1. Two related questions…
1. Determine the drag coefficient of a 75 kg skydiver with a projected area of 0.33 m2 and a terminal velocity of 60 m/s.
2. By how much would the skydiver need to reduce his project area so as to double his terminal velocity? How would he accomplish this?
2. Write something different.
3. Determine the velocity of a falling body as a function of time when…
1. drag is directly porpotional to speed and
2. drag is proportional to the square of speed.
This tab-delimited text file consists of just the power and top speed data for 122 cars tested by Road & Track magazine in 1998. Use the data in this file and your favorite analysis software to determine the model that best describes aerodynamic drag for automobiles; that is, determine the value of the power n in the generalized drag equation…

R = −bvn

### conceptual

1. A book and a page from that same book are both held horizontally and then dropped. Which one has…
1. the greater projected area (assuming no significant change in shape or orientation on the way down)?
2. the greater aerodynamic drag?
3. the greater weight?
4. the greater net force?
5. the greater velocity on impact with the floor?
2. A BASE jumper steps off the roof of a tall building followed shortly thereafter by a second jumper. The building is tall enough that aerodynamic drag should be considered. What happens to the separation between the two jumpers from the time the second jumper steps off the roof to the time when the first jumper lands on the ground?

### statistical

1. Two different bicycles were tested in a wind tunnel at the Massachusetts Institute of Technology (MIT) — an ordinary "stand up" road bike with drop handlebars and a recumbent bike (a bicycle you ride in a seated position). The drag force was measured at three different wind speeds while coasting and while pedaling. The rider on the road bike adopted three different postures. The recumbent was tested with and without a fairing (a plastic aerodynamic shield). Here are the measurements in their original English units.
Drag Force (lb) on Two Different Bicycles
speed (mph) without faring with faring on top bar on drops aero tuck
coast 10 1.92 2.40 2.40 1.44
20 10.08 8.64 10.56 7.20
30 24.00 19.68 26.88 15.84
pedal 10 2.88 3.12 2.64 3.12
20 9.36 9.84 10.56 10.08
30 24.48 19.44 26.40 20.40
Here are the same measurement in SI units
Drag Force (N) on Two Different Bicycles
speed (m/s) without faring with faring on top bar on drops aero tuck
coast 4.47 8.54 10.68 10.68 6.41
8.94 44.84 38.43 46.97 32.03
13.41 106.76 87.54 119.57 70.46
pedal 4.47 12.81 13.88 11.75 13.88
8.94 41.64 43.77 46.98 44.84
13.41 108.90 86.47 117.44 90.74
Determine the drag coefficient on the…
1. recumbent bike without a fairing
2. recumbent bike with a fairing
3. road bike with the rider's hands on the top bar of the handlebars
4. road bike with the rider's hands on the drops of the handlebars
5. road bike with the rider in the aero tuck posture
Source: Aerodynamic Performance of Vision Recumbents, Grant Bower, 01 November 1999.

### calculus

1. An object of mass m is thrown vertically upward with an initial speed +v0. The aerodynamic drag is directly proportional to the speed of the object. (Use b for the constant of proportionality.) State your solutions to the following problems in terms of m, v0, b, and fundamental constants.
1. Determine the following quantities (in whatever order you find most convenient) as functions of time from t = 0 until the object stops moving upward…
1. position
2. velocity
3. acceleration
2. How long does it take for the object to reach its maximum height?
3. To what maximum height does the object rise?
2. An object of mass m is thrown vertically downward with an initial speed +v0. The aerodynamic drag is directly proportional to the speed of the object. (Use b for the constant of proportionality.) State your solutions to the following problems in terms of m, v0, b, and fundamental constants.
1. Determine the following quantities (in any order you find convenient) as functions of time…
1. position
2. velocity
3. acceleration
2. Under what conditions will the acceleration…
1. always be positive?
2. always be negative?
3. start off positive and end up negative?
4. start off negative and end up positive?
3. A car of mass m is propelled forward starting from rest by a constant force +F. Use b for the constant of proportionality for all parts of this problem. State your solutions in terms of m, F, b, and fundamental constants.
1. Determine the terminal velocity of this car if it experiences…
1. a drag force directly proportional to speed.
2. a drag force proportional to the square of speed.
2. Determine the velocity as a function of time for this car if it experiences…
1. a drag force directly proportional to speed.
2. a drag force proportional to the square of speed.