To determine the height from which the package is released, we can analyze the motion of the package using the principles of projectile motion. Here's how:

1. Given Parameters:

   • Speed of the plane (horizontal velocity): $v$
   • Horizontal distance traveled by the package: $x$
   • Acceleration due to gravity: $g = 9.8 \, \text{m/s}^2$

2. Key Assumptions:

   • Air resistance is negligible.
   • The package is dropped, so it has no initial vertical velocity ($v_y = 0$).

3. Horizontal Motion: The horizontal motion of the package is governed by: $x = v \cdot t$ where $t$ is the time of flight.

4. Vertical Motion: The vertical motion of the package is governed by: $h = \frac{1}{2} g t^2$ where $h$ is the height from which the package is dropped.

5. Find Time of Flight: From the horizontal motion equation: $t = \frac{x}{v}$

6. Substitute $t$ into the Vertical Motion Equation: $h = \frac{1}{2} g \left(\frac{x}{v}\right)^2$ Simplify: $h = \frac{1}{2} g \frac{x^2}{v^2}$

Final Expression:

The height from which the package is released is: $h = \frac{g x^2}{2 v^2}$

Explanation:

This formula shows that the height depends on the horizontal distance ($x$), the speed of the plane ($v$), and the gravitational acceleration ($g$).

Would you like to see a step-by-step numerical example or have further clarifications?

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Related Questions:

1. How would air resistance change the result?
2. What happens to the height if the horizontal velocity doubles?
3. How can we calculate the time of flight for the package?
4. What additional factors would we consider for a real-world scenario?
5. How is the motion of the package affected if it is launched at an angle instead of being dropped?

Tip: Always double-check the units of your variables to ensure consistency when calculating motion parameters.