https://file.aiquickdraw.com/mathbot/file/1726274160072bdn31oar.jpg

The image provides a problem focused on verifying key equations in projectile motion: maximum height, flight time, and range.

### Key Equations:
1. **Maximum Height**: 
   \[
   y_{\text{max}} = \frac{{(v_0 \sin \alpha)^2}}{2g}
   \]
   
2. **Flight Time**: 
   \[
   t = \frac{{2v_0 \sin \alpha}}{g}
   \]
   
3. **Range**: 
   \[
   R = \frac{{v_0^2}}{g} \sin 2\alpha
   \]

---

### The Question:

To find the maximum height, you need to identify the correct position equations for **x** and **y** in terms of initial velocity (\(v_0\)) and angle (\(\alpha\)).

The correct answer selected in the image is:
- \( x = v_0 \cos \alpha \cdot t \)
- \( y = v_0 \sin \alpha \cdot t - \frac{1}{2} g t^2 \)

This option correctly describes the parametric equations for projectile motion where:
- The **x-component** is the horizontal motion, which is linear and depends on \( \cos \alpha \) and time.
- The **y-component** describes the vertical motion, which is influenced by gravity (hence the quadratic term \(\frac{1}{2}gt^2\)) and depends on \( \sin \alpha \).

### The Next Part:
In the bottom section, the problem asks for identifying the components of the vector position \(r\):
- The **x-component** is the *horizontal* position.
- The **y-component** is the *vertical* position.

Let me know if you would like further clarification or more details on how these formulas are derived.

### 5 Relative Questions:
1. How is the equation for maximum height derived from the y-component of the position function?
2. Can you explain the physical meaning of the flight time equation in projectile motion?
3. How does the angle \(\alpha\) affect the range of a projectile?
4. Why is the horizontal motion (x-component) independent of gravity?
5. What happens to the maximum height and range if the launch speed doubles?

### Tip:
When analyzing projectile motion, always remember that the horizontal and vertical motions are independent of each other, except for the time they share.

Verify the results for the maximum height, flight time, and range for ideal projectile motion. Identify the ideal motion equation from the choices provided.

Learn how to verify the results for maximum height, flight time, and range in ideal projectile motion. This guide covers parametric equations, key formulas, and trigonometric applications. Suitable for high school students (Grades 10-12).

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