# calculus problem velocity?

The velocity v of an object falling through the earth's atmosphere obeys the DE

dv/dt=g-kv

where k is called the drag constant and g is the gravitional constant. This equation states that the acceleration of the object is g reduced by an amount that represents air resistance(kv). Air resistance is proportional to the velocity of the object.

A find the function v(t) assuming that v(0)=0

B Let vT=g/k. When v=vT, the acceleration dv/dt is zero and thus the object maintains a constant terminal velocity. Rewirte v(t) in terms of vT and sketch the graph of v(t).

Update:

C For k=.2 how long does it take for an object to reach 95% of its terminal velocity?

Relevance

A)

dv/dt = g - kv

dv/(g - kv) = dt

dv/(g/k - v) = kdt

ln(g/k - v) = -kt + ln(C)

g/k - v = Ce^(-kt)

at v(0) = 0

g/k - 0 = Ce^(-k*0)

C = g/k

g/k - v = g/k*e^(-kt)

v(t) = g/k*(1 - e^(-kt))

B)

v(t) = vT*(1 - e^(-kt))

C)

v/vT = 1 - e^(-kt)

0.95 = 1 - e^(-0.2t)

e^(-0.2t) = 1 - 0.95

-0.2t = ln(0.05)

t = -ln(0.05)/0.2

t = 15.0 s