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AP Calculus Question of the Month: June
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by Ben Klein
Davidson College
Davidson, North Carolina
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1997 AB6, BC6: The Skydiver
Problem 6 on the 1997 AB and BC examinations was a common problem that dealt with the descent of a skydiver after her parachute opened. Students were given a differential equation and an initial condition and asked three interesting questions about the skydiver's descent.
We will propose two additional questions about the skydiver's descent, questions whose answers cast some doubt on the validity of the differential equation model given in the problem. We then suggest some ways in which the model might be modified to produce more realistic results, and we'll pursue one of them.
Remember that you can download solutions, scoring guidelines, and sample responses for the original problem right here on AP Central, but to make this Question of the Month self-contained, we will provide solutions for both the original parts of the problem and the new parts that appear below.
AB/BC6
Let 
be the velocity, in feet per second, of a skydiver at time 
seconds,
. After her parachute opens, her velocity satisfies the differential equation 
, with initial condition 
(0) = -50.
(A) Use separation of variables to find an expression for
in terms of
, where
is measured in seconds.
(B) Terminal velocity is defined as 
. Find the terminal velocity of the skydiver to the nearest foot per second.
(C) It is safe to land when her speed is 20 feet per second. At what time
does she reach this speed?
The three parts above appeared in the original version of the problem. The following parts extend them.
(D) If the skydiver's parachute opens at an altitude of 2,000 feet, how long will it take her to reach the ground, and will it be safe for her to land when she does reach the ground?
(E) What is the lowest possible altitude at which the skydiver's parachute could open and she would still reach the ground safely? [HINT: Think about your answer to part (C).]
You may find the answer to part (E) a little unrealistic. If so, you might want to change some of the parameters in the problem so as to make the number for the lowest possible altitude larger. The modifiable parameters are (1) the coefficient of
in the differential equation, (2) the value of
(0) = -50, and (3) the safe landing speed of 20 feet per second. The absolute value of the second parameter could be increased, but it would not be reasonable for the value to be much greater than 150 feet per second, which is about 100 miles per hour. The third parameter cannot be changed very much if we want the landing to be safe. This leaves us with the first parameter, the coefficient of
in the differential equation.
Thus, we will replace the differential equation and the initial condition with 
and
(0) = -150. Here 
is a positive parameter.
(F) Solve the new initial value problem and show that if the skydiver is to land safely,
must be at least 1.6.
(G) Find the smallest possible value of
such that the skydiver can land safely if her parachute opens at an altitude of 160 feet. With this value of
, how long will her descent take?
Click here to view the answers and commentary!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. He just completed a four-year term on the AP Calculus Development Committee.
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