Question

Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds. After her parachute opens, he velocity satisfies the differential equation dv/dt= -2v -32, with initial condition v(0)=-50
Use separation of variables to find an expression for v in terms of t.

Answer 1

Not sure if I did it right, but I got v(t)=-34e^(-2t)-16 ..?

Answer 2

@ganeshie8 @Loser66 could you help?

Answer 3

@agent0smith @campbell_st @jim_thompson5910 help?

Answer 4

Looks good !

Answer 5

Ok thanks! :)
Then what would I do about lim x->infinity of v(t) ?

Answer 6

take the limit and see

Answer 7

v(t)=-34e^(-2t)-16

Answer 8

I forgot how xD

Answer 9

as "t" increases,
the first term reduces right ?

Answer 10

cuz its a negative exponent

Answer 11

as \(t \to \infty\), \(e^{-t} = \dfrac{1}{e^t} \to ?\)

Answer 12

Right ... So the -34e^-2t part approaches 0?
So I get -16?

Answer 13

exactly ! the first term vanishes completely, and the velocity approaches -16

Answer 14

Great! :D
Thank you! :)

Answer 15

np :)

Answer 16

if you're hearing it for the first time, its an amazing result actually

Answer 17

the free falling body cannot continue to increase its velocity forever - the velocity saturates after reaching a certain limit

Answer 18

google `terminal velocity` for more :)

Answer 19

Makes sense or you could be falling at speed of infinity xD
Then "It is safe to land when her speed is 20 ft per sec. At what time does she reach this speed?"

Answer 20

Would that be 20= -34e^-2t -16 ?

Answer 21

v(t)=-34e^(-2t)-16


-20 =-34e^(-2t)-16

solve t

Answer 22

yeah but wolfram seems to be confused...
http://www.wolframalpha.com/input/?i=20+%3D+-34e%5E%28-2x%29+-16

Answer 23

the velocity is negative, cuz you're falling down, and the given differential equation assumes that the increasing position is UP.

Answer 24

I'm confused? :/

Answer 25

so, velocity is POSITIVE, if u are going in that direction.
velocity is NEGATIVE, if u are going opposite to that direction

Answer 26

She's falling up? O.o

Answer 27

since the given differential equation assumes increasing height is toward UP (why ?),
and since she is falling down, that means she is going in opposite direction to the increasing height. so velocity is NEGATIVE.

Answer 28

Erm... So whut do i do?

Answer 29

set ur solution equal to -20 and solve t

Answer 30

t=1.07 ...?

Answer 31

thats right ! after 1.07 seconds her speed reaches 20 units/sec

Answer 32

Alright! :) Thank you for your help! <3

Answer 33

you wlcme :)