The position of an object moving along a line is given by the function st) = - 16? + 96t. Find the average
velocity of the object over the following intervals.
(a) [1, 6] (c) [1, 4]
(b) [1, 5]
(d) [1, 1 + h] where h > 0 is any real number.

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 \(s(t)\).

The position of an object moving along a line is given by the function s(t) = -16t^2 + 96t. Find the average velocity of the object over the following intervals: (a) [1, 6], (b) [1, 5], (c) [1, 4], (d) [1, 1 + h] where h > 0 is any real number.

Explore how to calculate the average velocity of an object using the position function s(t) = -16t^2 + 96t over various intervals. This problem covers essential calculus concepts and is suitable for students in Grades 11-12.

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