The motion of a body falling from rest in a resisting medium is described by the equation `(dv)/(dt)=a-bv` where a and b are constant . The velocity at any time `t` is given by

Options

A. `v=(a)/(b)(1-e^(-bt))`
B. `v=(b)/(a)(e^(-bt))`
C. `v=(a)/(b)(1+e^(-bt))`
D. `v=(b)/(a)e^(bt)`

Answer

A

Explanation

`(dv)/(dt)=a-bvint_(0)^(v)(dv)/(a-bv)=int_(0)^(t)dt`
`((-2)/(b))[`in `(a-bv)]_(0)^(v)=timplies "ln"(a-bv)/(a)=-bt`
`impliesa-bv=ae^(-bt)`
`impliesv=(a)/(b)(1-e^(-bt))`