an aircraft flying at 550 feet over level terrain drops a supply package. What equation would you use to set up the mathematical model for the height of the package. explain why

To model the height of a supply package dropped from an aircraft flying at 550 feet, we would use the **equation of motion** for free fall under the influence of gravity.

The equation is:

\[
h(t) = h_0 - \frac{1}{2} g t^2
\]

### Explanation:

- **\( h(t) \)** is the height of the package at time \( t \).
- **\( h_0 = 550 \, \text{feet} \)** is the initial height of the package (the altitude of the aircraft).
- **\( g \approx 32 \, \text{ft/s}^2 \)** is the acceleration due to gravity near the Earth's surface (in feet per second squared).
- **\( t \)** is the time in seconds after the package is dropped.
- The negative sign indicates that the height is decreasing as the package falls.

### Why this equation?
This equation is derived from basic kinematics and assumes no air resistance (ideal free fall). The term \( \frac{1}{2} g t^2 \) represents the distance fallen due to gravity over time \( t \). Subtracting this distance from the initial height gives us the current height of the package at any given time.

Would you like more details or have any questions?

Here are some related questions:
1. How would the equation change if the package is dropped from a different height?
2. What is the time it takes for the package to hit the ground?
3. How does air resistance affect the model in real-world scenarios?
4. How would you model the horizontal distance traveled by the package if the aircraft is moving?
5. How would you modify the model for a package dropped from a moving aircraft in a windy environment?

**Tip:** To find when the package hits the ground, set \( h(t) = 0 \) and solve for \( t \).

An aircraft flying at 550 feet over level terrain drops a supply package. What equation would you use to set up the mathematical model for the height of the package? Explain why.

Learn how to model the height of a package dropped from an aircraft flying at 550 feet using the free fall equation of motion. This problem explores kinematics and gravity concepts and is suitable for students in Grades 9-12. Key formula: h(t) = h0 - (1/2) * g * t^2.

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