Answer:
[tex]v(t)=\dfrac{a_o}{b}[{1-e^{-bt}][/tex]
Explanation:
The acceleration of an object decreases exponentially with time as :
[tex]a(t)=a_oe^{-bt}[/tex]
We know that, the relation between the velocity and the acceleration is given by :
[tex]v(t)=\int\limits{a(t).dt}[/tex]
Put the value of a(t) in above equation. So,
[tex]v(t)=\int\limits{(a_oe^{-bt}).dt}[/tex]
[tex]v(t)=a_o\int\limits{(e^{-bt}).dt}[/tex]
[tex]v(t)=\dfrac{-a_oe^{-bt}}{b}} +c[/tex]
At t = 0, v(t) = 0
So, [tex]0=\dfrac{-a_oe^{-b(0)}}{b}} +c[/tex]
[tex]k=\dfrac{a_o}{b}[/tex]
So, its velocity is given by :
[tex]v(t)=\dfrac{-a_oe^{-bt}}{b}} +\dfrac{a_o}{b}[/tex]
[tex]v(t)=\dfrac{a_o}{b}[{1-e^{-bt}][/tex]
Hence, this is the required solution.
