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. `(A)/(B) (1 - e^(B t))`
B. `A (1 - e^(-B^(2)t))`
C. `Abe^(-1)`
D. `AB^(2) (1 - t)`

Answer

A

Explanation

`(dv)/(dt) = (A - Bv)`
`underset(0)overset(v)int (dv)/((A - Bv)) = underset(0)overset(t)int dt`
`-(1)/(B) log_(e) (A - Bv) underset(0)overset(v):| = t :|_(0)^(t)`
`-(1)/(B) [log_(e) (A - Bv) - log_(e) A] = t :|_(0)(t)`
`log_(e) ((A - Bv)/(A)) = - Bt`
`(A - Bv) = Ae^(-Bt)`
`Bv = A - Ae^(-Bt)`
`v = (A)/(B) (1 - e^(-Bt))`