https://file.aiquickdraw.com/mathbot/file/1726086655203cdpr5f7b.jpg

The problem involves finding the velocity of a free-falling object at a specific time using the given position function:

\[
s(t) = -4.9t^2 + 250
\]

where \( s(t) \) is the height in meters of the object after \( t \) seconds. The velocity of the object at time \( t = a \) seconds is given by the limit:

\[
\lim_{{a \to t}} \frac{s(a) - s(t)}{a - t}
\]

This is the definition of the derivative of the function \( s(t) \), which represents the instantaneous velocity of the object at time \( t \).

To find the velocity of the object when \( t = 4 \), we need to compute the derivative of \( s(t) \) and evaluate it at \( t = 4 \).

### Step 1: Differentiate the position function

The derivative of \( s(t) = -4.9t^2 + 250 \) with respect to \( t \) is:

\[
s'(t) = \frac{d}{dt}(-4.9t^2 + 250) = -9.8t
\]

### Step 2: Evaluate the derivative at \( t = 4 \)

Now, substitute \( t = 4 \) into the derivative to find the velocity:

\[
s'(4) = -9.8 \cdot 4 = -39.2
\]

### Conclusion

The velocity of the object at \( t = 4 \) seconds is \(-39.2\) meters per second. The negative sign indicates that the object is moving downward.

Would you like more details, or do you have any questions about this solution?

Here are some related questions you might find interesting:

1. What does the derivative of a position function represent in general?
2. How would the velocity change if we changed the height or acceleration due to gravity?
3. What is the significance of the negative velocity in this context?
4. How would you find the velocity at a different time, such as \( t = 6 \)?
5. What is the object's acceleration, and how is it related to the velocity function?

**Tip:** When solving problems involving free-falling objects, remember that gravity causes constant acceleration, which is reflected in the quadratic term of the position function.

Find the velocity of a free-falling object at t = 4 using the position function s(t) = -4.9t^2 + 250.

Learn how to find the velocity of a free-falling object using the position function s(t) = -4.9t^2 + 250 and differentiation. This detailed explanation covers the step-by-step process of finding the derivative and evaluating it at t = 4 seconds.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation