Velocity of skydiver satisfies dv/dt = -2v - 32, v(0) = -50

[Undeuxtroiscatsank?]

Hi, I'm new here! I'm hoping someone can help me with this problem. Thanks. So, anyways...

1.) 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 equation dv/dt = -2v - 32, with initial condition v(0) = -50.

a) Find an equation for v(t).

I'm guessing I have to take the integral of "-2v - 32" for this one. I'm not sure, though. Please help me start it off, or rough directions I should follow, if possible.

b) Termnial Velocity is defined as limit t →∞ v(t). Find the terminal velocity of the skydiver to the nearest foot per second.

I think I have to know how to do part a) do part b).

c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?

?

Thank you so much. Any suggestions are appreciated.

 

[stapel]

a) Integrate dv/dt with respect to t, using the initial condition to find the value of the integration constant.

b) Take the limit as t gets infinitely large.

c) Set v(t) equal to "20" and solve for the time t.

If you get stuck, please reply showing all of your work and reasoning. Thank you!

Eliz.

 

[skeeter]

Re: Velocity of skydiver satisfies dv/dt = -2v - 32, v(0) =

Hi, I'm new here! I'm hoping someone can help me with this problem. Thanks. So, anyways...

1.) 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 equation dv/dt = -2v - 32, with initial condition v(0) = -50.

a) Find an equation for v(t).

I'm guessing I have to take the integral of "-2v - 32" for this one. I'm not sure, though. Please help me start it off, or rough directions I should follow, if possible.

dv/dt = -2v - 32
dv/dt = -2(v + 16)
seperate variables ...
dv/(v+16) = -2 dt
now integrate.

b) Termnial Velocity is defined as limit t →∞ v(t). Find the terminal velocity of the skydiver to the nearest foot per second.

I think I have to know how to do part a) do part b).

correct

c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?

you guessed it ... you need v as a function of time to answer this one too. remember that speed = |v|.

?

Thank you so much. Any suggestions are appreciated.

 

[Undeuxtroiscatsank?]

Wow! Thanks for your replies! So, this is what I've done:
a)
∫ dv/(v+16) = ∫ -2dt
ln |v+16| = -2t + C
v+16 = e^(-2t + C)
v+16 = (e^-2t) · C
v = (e^-2t) - 16 · C
Initial condition v(0)= -50

-50 = e^(-2(0)) - 16 · C
-50 = 1 - 16 · C
-35 = C
v = e^(-2t) - 16 · (-35)

Is this correct?

And for part (b) I take the limit of "e^(-2t) - 16 · (-35)", right? How would that work out?

For (c), I think I messed up:

v = e^(-2t) - 16 · (-35)
20= e^(-2t) - 16 · (-35)
20= e^(-2t) · 560
20/560 = e^(-2t)
ln 20/560 = -2t
ln 20 - ln 560 = -2t
2.996 - 6.328 = -2t
-3.332 = -2t
-3.332/-2 = t
1.666 = t



Thank you for all your help! I still can't get over the fact that you take the time to look at my questions. You've been great help!