https://file.aiquickdraw.com/mathbot/file/17273057498032jjkmd69.jpg

To solve the problem, we are given the function that represents the height \( s(x) \) of an object above the ground after \( x \) seconds, where:

\[
s(x) = 1936 - 16x^2
\]

We need to find two things:
1. The time it takes for the object to hit the ground.
2. The object's velocity at the moment of impact.

### 1. Time to Hit the Ground
The object hits the ground when its height \( s(x) \) becomes zero. Therefore, we set the equation equal to zero and solve for \( x \):

\[
1936 - 16x^2 = 0
\]

\[
16x^2 = 1936
\]

\[
x^2 = \frac{1936}{16}
\]

\[
x^2 = 121
\]

\[
x = \sqrt{121} = 11 \text{ seconds}
\]

So, the object takes **11 seconds** to hit the ground.

### 2. Velocity at the Moment of Impact
The velocity \( v(x) \) of the object is the derivative of the height function \( s(x) \) with respect to time \( x \). 

\[
v(x) = \frac{ds(x)}{dx} = \frac{d}{dx}\left(1936 - 16x^2\right) = -32x
\]

To find the velocity at the moment of impact, substitute \( x = 11 \) seconds into the velocity equation:

\[
v(11) = -32(11) = -352 \text{ ft/s}
\]

The negative sign indicates the direction of the velocity (downward).

Therefore, the velocity of the object at the moment of impact is **352 ft/s downward**.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How would the time change if the object were dropped from a different height?
2. What would be the impact velocity if the height function had a different coefficient for \( x^2 \)?
3. How would air resistance alter the time to hit the ground and the impact velocity?
4. Can we use the same method to determine the time of impact for an object thrown upwards?
5. What would the height function look like if the object had an initial upward velocity?

**Tip:** When dealing with free-fall problems, always consider the direction of motion; downward motion is typically considered negative in the velocity.

An object is dropped from a tower, 1936 ft above the ground. The object's height above ground x seconds after the fall is s(x) = 1936 - 16x^2. About how long does it take the object to hit the ground? What is the object's velocity at the moment of impact?

Solve a free fall problem where an object is dropped from 1936 feet above the ground. Learn how to calculate the time it takes to hit the ground and the velocity at the moment of impact using quadratic equations and derivatives. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation