Question

If the acceleration of an object is given by dv/dt = −2v, Find the position function s(t) if v(0) = 7 and s(0) = 0.

Answer 1

Explanation:

Given that acceleration of an object is

`(dv)/(dt) =-2v\\(dv)/(v) =-2dt\\ln v = -2t+C\\`

is the solution to the differential equation

Since v(0) =7

we get ln 7 = C

Hence
`lnv = -2t+ln 7\\v=7e^(-2t)`

since velocity is rate of change of distance s we have

`v=(ds)/(dt) =7e^(-2t)\\s= [tex]s(t) =(-7)/(2) (e^(-2t))+C)[`

substitute t=0 and s=0

`C=7/2`

So solution for distance is

`s(t) =(-7)/(2) (e^(-2t)-1)`