# Aerodynamic Drag

## Problems

### practice

- Two related questions…
- Determine the drag coefficient of a 75 kg skydiver with a projected area of 0.33 m
^{2}and a terminal velocity of 60 m/s. - By how much would the skydiver need to reduce her project area so as to double her terminal velocity? How would she accomplish this?

- Determine the drag coefficient of a 75 kg skydiver with a projected area of 0.33 m
- The 17th century French scientist Edme Mariotte was the first to realize that aerodynamic drag is proportional to the square of speed. Predicting the position of a freely falling body was made possible by Galileo Galilei only half a century earlier. Mariotte took it one step further and predicted the speed of an object falling under the influence of gravity
*and*air resistance.Voici des Tables faites sur cette hypothèse, par lesquelles on connoîtra combien une balle de plomb de six lignes de diamètre passera de pieds en chaque seconde en descendant; combien elle en passera dans tel nombre de secondes qu'on voudra choisir; quand elle cessera d'accélérer son mouvement; quelle sera sa vites se complette; & combien elle parcourra de pieds avant que de l'acquérir.

These tables, made by applying this hypothesis, show how many feet a lead ball six lines [1.3535 cm] in diameter will fall in each second; how many feet it will fall in any number of seconds we choose; when and where it stops accelerating; what will be its final speed; & how many feet it will cover before acquiring it.

Adapted from Edme Mariotte, 1673

*Ancien Régime*units to metric ones for you.*Pieds*are the French equivalent of, and very similar to, English feet.*Lignes*(lines) are subdivisions of a*pied*. 144*lignes*make a pied. 443.296*lignes*equal a meter.- Construct a graph of distance vs. time from Mariotte's predicted values and use it to determine the terminal velocity of his hypothetical lead ball.
- Use the contemporary drag equation,
*R*= ½ρ*CAv*^{2}, that evolved from Mariotte's hypothesis to determine the terminal velocity of his hypothetical lead ball. - How do the results of your two analyses compare? Did Monsieur Mariotte make a good prediction? Keep in mind that he had no way to test it. The stopwatch was two centuries away, at least.

- Determine the velocity of a falling body as a function of time when…
- drag is directly porpotional to speed and
- drag is proportional to the square of speed.

- road-test-summary.txt

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*= −*bv*^{n}

### conceptual

- A book and a page from that same book are both held horizontally and then dropped. Which one has…
- the greater projected area (assuming no significant change in shape or orientation on the way down)?
- the greater aerodynamic drag?
- the greater weight?
- the greater net force?
- the greater velocity on impact with the floor?

- 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

- 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 recumbent road bike 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 Drag force (N) on two different bicycles recumbent road bike 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 - recumbent bike without a fairing
- recumbent bike with a fairing
- road bike with the rider's hands on the top bar of the handlebars
- road bike with the rider's hands on the drops of the handlebars
- road bike with the rider in the aero tuck posture

- cyclists-in-a-wind-tunnel.txt

Aerodynamic drag is the greatest source of resistance to cyclists riding on level ground. To better understand the effects of posture, clothing, and equipment on drag, cyclists and their gear are often placed in wind tunnels where conditions are easier to control. The table below, and in the accompanying text file, show data collected for eight competition cyclists in a wind tunnel. The riders were seated on time trial bicycles and adopted a standard aero racing posture. Their heights and masses without the bicycle were measured in the usual way for a medical exam. Drag forces were measured with the wind tunnel set to 48 km/h. Their projected areas were measured photographically before entering the wind tunnel using the same posture and bicycle. Use the data given to complete the table.Aerodynamic data for eight cyclists in a wind tunnel Source: Bassett, et al. 1999 as cited in García‑López, et al. 2008. cyclist number height (m) mass (kg) drag force (N) projected area (m ^{2})drag coefficient 1 1.63 47.6 23.00 0.212 2 1.75 59.9 23.22 0.214 3 1.80 69.0 24.99 0.230 4 1.80 74.0 21.01 0.194 5 1.80 74.0 20.42 0.188 6 1.80 77.0 21.35 0.197 7 1.86 81.0 20.42 0.187 8 1.93 87.0 22.79 0.210 mean → standard deviation →

### calculus

- An object of mass
*m*is thrown vertically*upward*with an initial speed +*v*_{0}. 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*,*v*_{0},*b*, and physical constants.- Determine the following quantities (in whatever order you find most convenient) as functions of time from
*t*= 0 until the object stops moving upward…- position
- velocity
- acceleration

- How long does it take for the object to reach its maximum height?
- To what maximum height does the object rise?

- Determine the following quantities (in whatever order you find most convenient) as functions of time from
- An object of mass
*m*is thrown vertically*downward*with an initial speed +*v*_{0}. 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*,*v*_{0},*b*, and physical constants.- Determine the following quantities (in any order you find convenient) as functions of time…
- position
- velocity
- acceleration

- Under what conditions will the acceleration…
- always be positive?
- always be negative?
- start off positive and end up negative?
- start off negative and end up positive?

- Determine the following quantities (in any order you find convenient) as functions of time…
- 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 physical constants.- Determine the terminal velocity of this car if it experiences…
- a drag force directly proportional to speed.
- a drag force proportional to the square of speed.

- Determine the velocity as a function of time for this car if it experiences…
- a drag force directly proportional to speed.
- a drag force proportional to the square of speed.

- Determine the terminal velocity of this car if it experiences…

### investigative

- What is the maximum speed of a car?
- First derive an equation for the maximum speed of a car driving on level ground with an air resistance given by the equation…

State your answer using only the symbols in the equation above and the power,*R*= ½ρ*CAv*^{2}*P*, of the car. - Use the equation you just derived to compute a "good enough" answer. Assume a spherical car with a radius of 1 m and a power of 100 kW.
- Now compute a "more realistic" answer using values for
*P*,*C*, and*A*that correspond to an actual, non-spherical car. Be sure to state the make, model, production year, and source of this information.

- First derive an equation for the maximum speed of a car driving on level ground with an air resistance given by the equation…