  ### Home > CALC > Chapter 3 > Lesson 3.2.3 > Problem3-78

3-78.

The position of a ball as a function of time is given by the function below where $s(t)$ is in meters and $t$ is in seconds.

$s ( t ) = \sqrt { t + 1 }$

1. Use your calculator to approximate the instantaneous velocity of the ball at $1, 5, 10,$ and $100$ seconds.

Instantaneous velocity = Instantaneous Rate of Change (IROC) = Derivative

$\text{Rewrite }s(t)\text{ using exponents: }s(t)=(t+1)^{1/2}$

Find $s'(t)$ using the Power rule.

Evaluate $s'(1)$, $s'(5)$, $s'(10)$ and $s.(100)$.

2. What do you predict happens to the velocity of the ball after a very long time (i.e. as $t→∞$)?

Note: $t = 100$ seconds is a very long time for a ball to be in motion. (This must be an unearthly situation!) Is $s(t)$ increasing, decreasing or neither as $t→∞$? Is $s(t)$ changing rapidly or not so rapidly? Explain.

3. What happens to the position of the ball after a very long time, (i.e. what is $s(t)$)? Does this make sense given your answer to part (b)?

Does a square root function (such as $s(t)$) have a horizontal asymptote as $t→∞$?