Lecture Notes in Differential Equations - Bruce E. Shapiro

51
this becomes a first-order ODE in v
m dv
dt = −mg − C Dv (6.51)
Example 6.5. Suppose we drop a 2010 penny off the top of the empire
state building. How long will it take to hit the ground?
The drag coefficient of a penny is approximately C D ≈ 1 and its mass is
approximately 2.5 grams or 0.0025 kilograms. The height of the observation
deck is 1250 feet = 381 meters.
Since the initial velocity is zero, the IVP is
}
v ′ = −9.8 − v
v(0) = 0
(6.52)
Rearranging,
An integrating factor is e t so
v ′ + v = −9.8 (6.53)
Integrating
Dividing by e t ,
From the initial condition,
d (
ve
t ) = −9.8e t (6.54)
dt
ve t = −9.8e t + C (6.55)
v ≈ −9.8 + Ce −t (6.56)
hence
0 = −9.8 + C =⇒ C = 9.8 (6.57)
v = −9.8 + 9.8e −t (6.58)
Now we go back to our substitution of (6.50), we have another first order
linear differential equation:
Integrating (6.59)
dy
dt = −9.8 + 9.8e−t (6.59)
y = −9.8t − 9.8e −t + C (6.60)
If the object is dropped from a height of 381 meters then
y(0) = 381 (6.61)