# More Differential Eqns

• May 4th 2009, 06:47 PM
ny_chow
More Differential Eqns
Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t>0. After her parachute opens, her velocity satisfies the differential eqn
dv/dt = -2v-32,
with initial condition v(0) = -50

b) Terminal velocity is defined as limv(t) as t approaches infinity. 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/second. At what time t does she reach this speed?
• May 4th 2009, 07:00 PM
TheEmptySet
Quote:

Originally Posted by ny_chow
Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t>0. After her parachute opens, her velocity satisfies the differential eqn
dv/dt = -2v-32,

with initial condition v(0) = -50

b) Terminal velocity is defined as limv(t) as t approaches infinity. 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/second. At what time t does she reach this speed?

This equation is seperatable

$\frac{dv}{dt}=-2(v+16) \iff \frac{dv}{v+16}=-2dt$

Integrating both sides we get

$\ln|v+16|=-2t+c \iff v+16 =e^{-2t+c} \iff v(t)=Ae^{-2t}-16$

Now we use $v(0)=-50=Ae^{0}-16 \iff A=-34$

$v(t)=-34e^{-2t}-16$

You should be able to finish from here