Find the time that must elapse for the object to reach 98% of its limiting velocity? I am given the initial value problem $$ \frac{dv}{dt} = 9.8 - (\frac v5) $$ and you are given $v(0) = 0$ I was looking at the solution to this problem. They first solved the differential equation. They got $v(t) = 49(1-e^-{\frac t5})$I completely understand that process. Then, they use this equation $v(T) = 0.98v(t_{\infty})$ I don't understand where they got this equation or what it even means. Can someone explain this equation to me.

The limiting velocity is $\lim_{t->\infty} v$, denoted by $v(t_{\infty})$. Then $98\%$ of the limiting velocity is $98\% v(t_{\infty})$. So you are just trying to find $T$ such that $v(T)=98\% v(t_{\infty})$. This is in fact where you will start with. It didn't solve anything yet.

You will need to use the above solution of the ODE in this equation to solve for $T$.