where v is the velocity in meters per second Find the time i

where v is the velocity in meters per second. Find the time, in seconds, that must elapse for the object to reach 96% of its limiting velocity. How far, in meters, does the object fall in that time?

Solution

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A falling object satisfies the initial value problem
dv/dt = 9.8 (v/5), v(0) = 0.
A) Find the time that must elapse for the object to reach 98% of its limiting velocity.
B) How far does the object fall in the time found in part A?

A) dv/dt = (49 - v) / 5
(1/5)dt = dv / (49 - v)
t / 5 + C= -ln(49 - v)
0 + C = -ln(49)
C = -ln(49)
t/5 - ln(49) = -ln(49 - v)
ln(49) - t/5 = ln(49 - v)
49 - v = e^(ln(49) - t/5) = 49 e^(-t/5)
v = 49 - 49e^(-t/5)

As t goes to infinity, v goes to 49, so that\'s the limiting speed.
98% of 49 is 48.02.

48.02 = 49 - 49e^(-t/5)
0.98 = 49 e^(-t/5)
0.02 = e^(-t/5)
-t/5 = -3.91
t = 19.56