A falling object does not keep accelerating indefinitely but, due to air resistance, reaches a terminal speed. What is the terminal speed?

Question

Suppose that the speed of such an object, t seconds after the fall commences is vm/s where v=

$$\frac{200}{3}(1-e^{-0.15t})$$

Find the speed of the object after five seconds.

I have substituted t=5, getting 35.2km/h

What is the terminal speed?

I know the answer is: $$\frac{200}{3}m/s$$

But is there a formula to calculate this/what is the logic to getting this answer?

This is from a Year 12 Methods textbook.

Answer 1

Since the object in free fall will reach terminal velocity at the very end, you have to take the limit of the velocity as $t$ approaches infinity.

$$v=\frac{200}{3}(1-e^{-0.15t})$$

As $t$ grows till $\infty$, the value of $e^{-kx}$ approaches zero. Hence we can substitute $t=0$ in our equation. Hence we get- $$v_{terminal}=\frac{200}3$$