Question

By comparing the surface area of a sphere with its volume and assuming that air resistance is proportional to the square of velocity, it is possible to make a heuristic argument to support the following premise: For similarly shaped objects, terminal velocity varies in proportion to the square root of length. Expressed in a formula, this is

T = kL0.5,
where L is length, T is terminal velocity, and k is a constant that depends on shape, among other things. This relation can be used to help explain why small mammals easily survive falls that would seriously injure or kill a human. (Round your answers to two decimal places.)
(a) A 6-foot man is 36 times as long as a 2-inch mouse (neglecting the tail). How does the terminal velocity of a mancompare with that of a mouse?
The terminal velocity of the man is times that of the mouse.
(b) If the 6-foot man has a terminal velocity of 120 miles per hour, what is the terminal velocity of the 2-inch mouse?
mph
(c) Neglecting the tail, a squirrel is about 8 inches long. Again assuming that a 6-foot man has a terminal velocity of 120miles per hour, what is the terminal velocity of a squirrel?
mph

Answer 1

(a) To compare the terminal velocity of a man with that of a mouse, we need to use the given formula:
T = kL^0.5
where L is length, T is terminal velocity, and k is a constant that depends on shape, among other things.
Let T_m and L_m be the terminal velocity and length of the man, and T_r and L_r be the terminal velocity and length of the mouse. Then we have:
T_m = kL_m^0.5 T_r = kL_r^0.5
Dividing the first equation by the second equation, we get:
T_m/T_r = (L_m/L_r)^0.5
Since L_m is 36 times as long as L_r, we have:
T_m/T_r = (36)^0.5 T_m/T_r = 6
Therefore,
the terminal velocity of the man is 6 times that of the mouse.
(b) To find the terminal velocity of the mouse, we need to use the given formula again:
T_r = kL_r^0.5
where L_r is the length of the mouse, which is 2 inches or 1/6 feet. We also know that T_m is 120 miles per hour and L_m is 6 feet. So we have:
T_m = kL_m^0.5 120 = k(6)^0.5
Solving for k, we get:
k = 120/(6)^0.5 k = 49.01
Substituting this value of k into the formula for T_r, we get:
T_r = 49.01(1/6)^0.5 T_r = 20
Therefore,
the terminal velocity of the 2-inch mouse is 20 miles per hour.
© To find the terminal velocity of a squirrel, we need to use the given formula again:
T_s = kL_s^0.5
where L_s is the length of the squirrel, which is 8 inches or 2/3 feet. We also know that k is 49.01 from part (b). So we have:
T_s = 49.01(2/3)^0.5 T_s = 40
Therefore,
the terminal velocity of an 8-inch squirrel is 40 miles per hour.