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

Answer:

Step-by-step explanation:

Given that acceleration of an object is

$\frac{dv}{dt} =-2v\\\frac{dv}{v} =-2dt\\ln v = -2t+C$

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

$v=\frac{ds}{dt} =7e^{-2t}\\s= [tex]s(t) =\frac{-7}{2} (e^{-2t})+C)[$

substitute t=0 and s=0

$C=7/2$

So solution for distance is

$s(t) =\frac{-7}{2} (e^{-2t}-1)$