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The problem requires finding the average velocity of an object moving vertically along a line, given its position function \( s(t) = -16t^2 + 96t \) over the interval from \( t = 1 \) to \( t = 4 \). The average velocity \( \overline{v} \) can be calculated using the formula:

\[
\overline{v} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]

Plugging in the values from the position function, where \( t_1 = 1 \) and \( t_2 = 4 \), we calculate:

1. Calculate \( s(1) \):
\[
s(1) = -16(1)^2 + 96(1) = -16 + 96 = 80 \, \text{meters}
\]

2. Calculate \( s(4) \):
\[
s(4) = -16(4)^2 + 96(4) = -16 \times 16 + 384 = -256 + 384 = 128 \, \text{meters}
\]

Now, find the average velocity:
\[
\overline{v} = \frac{128 - 80}{4 - 1} = \frac{48}{3} = 16 \, \text{m/s}
\]

Therefore, the average velocity of the object over the interval from \( t = 1 \) to \( t = 4 \) is \( 16 \, \text{m/s} \), which corresponds to answer choice \([C] 16 \, \text{m/s}\).

Would you like further details on this problem, or do you have any other questions? Here are some related questions to consider:

1. What is the instantaneous velocity of the object at \( t = 1 \)?
2. What is the instantaneous velocity of the object at \( t = 4 \)?
3. How would the average velocity change if the interval was from \( t = 0 \) to \( t = 4 \)?
4. Can you calculate the maximum height reached by the object?
5. What is the acceleration of the object at any time \( t \)?

**Tip**: When solving for average velocity over a time interval, always use the position values at the endpoints of the interval and divide by the difference in time to get the average rate of change.

The position of an object moving vertically along a line is given by s(t) = -16t^2 + 96t. Find the average velocity of the object over the interval [1,4] if s is in meters and t is in seconds.

This problem demonstrates how to find the average velocity of an object moving in a vertical line with a position function s(t) = -16t^2 + 96t. We calculate it over the time interval from 1 to 4 seconds, using basic calculus concepts of rate of change.

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