To find the average velocity of an object over a specific interval, we use the formula:

$\text{Average velocity over } [a, b] = \frac{s(b) - s(a)}{b - a}$

where $s(t)$ is the position function of the object.

Given the position function:

$s(t) = -16t^2 + 96t$

we'll compute the average velocity for each interval.

(a) Interval $[1, 6]$

Substitute $t = 1$ and $t = 6$ into the position function:

$s(1) = -16(1)^2 + 96(1) = -16 + 96 = 80$

$s(6) = -16(6)^2 + 96(6) = -16(36) + 576 = -576 + 576 = 0$

The average velocity over the interval $[1, 6]$ is:

$\text{Average velocity} = \frac{s(6) - s(1)}{6 - 1} = \frac{0 - 80}{5} = \frac{-80}{5} = -16$

(b) Interval $[1, 5]$

Substitute $t = 1$ and $t = 5$ into the position function:

$s(5) = -16(5)^2 + 96(5) = -16(25) + 480 = -400 + 480 = 80$

The average velocity over the interval $[1, 5]$ is:

$\text{Average velocity} = \frac{s(5) - s(1)}{5 - 1} = \frac{80 - 80}{4} = \frac{0}{4} = 0$

(c) Interval $[1, 4]$

Substitute $t = 1$ and $t = 4$ into the position function:

$s(4) = -16(4)^2 + 96(4) = -16(16) + 384 = -256 + 384 = 128$

The average velocity over the interval $[1, 4]$ is:

$\text{Average velocity} = \frac{s(4) - s(1)}{4 - 1} = \frac{128 - 80}{3} = \frac{48}{3} = 16$

(d) Interval $[1, 1 + h]$ where $h > 0$

To find the average velocity over the interval $[1, 1 + h]$, we use:

$\text{Average velocity} = \frac{s(1 + h) - s(1)}{(1 + h) - 1} = \frac{s(1 + h) - s(1)}{h}$

First, compute $s(1 + h)$:

$s(1 + h) = -16(1 + h)^2 + 96(1 + h)$

Expand and simplify:

$s(1 + h) = -16(1 + 2h + h^2) + 96 + 96h = (-16 - 32h - 16h^2) + 96 + 96h$

$s(1 + h) = -16h^2 + 64h + 80$

Now, compute the difference:

$s(1 + h) - s(1) = (-16h^2 + 64h + 80) - 80 = -16h^2 + 64h$

Thus, the average velocity is:

$\text{Average velocity} = \frac{-16h^2 + 64h}{h} = -16h + 64$

Summary of Results:

(a) Average velocity over $[1, 6]$ = $-16$

(b) Average velocity over $[1, 5]$ = $0$

(c) Average velocity over $[1, 4]$ = $16$

(d) Average velocity over $[1, 1 + h]$ = $-16h + 64$

Would you like more details or have any other questions?

Related Questions:

1. What is the instantaneous velocity of the object at $t = 2$?
2. How does the average velocity change as $h$ approaches 0 for interval $[1, 1 + h]$?
3. What is the acceleration of the object at any time $t$?
4. What is the significance of negative average velocity?
5. Can you interpret the motion of the object using the position function?

Tip: The instantaneous velocity is the derivative of the position function $s(t)$.