https://file.aiquickdraw.com/mathbot/file/17273240458854wvxou4b.jpg

Let's analyze both problems presented in the image.

### Problem 1: Football Kick
- **Given**: A football was kicked 126 feet, and the time of flight was 3.6 seconds.
- **Objective**: Determine the initial speed \( v_0 \) and the angle \( \theta \) at which it was kicked.
  
For projectile motion, the horizontal range \( R \), time of flight \( t \), and initial velocity components are related by the equations:
\[
R = v_0 \cdot \cos(\theta) \cdot t
\]
\[
h(t) = v_0 \cdot \sin(\theta) \cdot t - \frac{1}{2} g t^2
\]
where \( g = 32.2 \, \text{ft/s}^2 \) (gravitational acceleration in feet per second squared). Since the total horizontal range is given, and the time of flight is known, these two equations can be solved for \( v_0 \) and \( \theta \).

### Problem 2: Basketball Shot
- **Given**: The basketball shot passes just over player B's hand at height \( h \) and through the hoop. The horizontal distances are shown in the figure, with the initial height 7 feet, the hoop 10 feet high, and the angle \( \theta = 30^\circ \).
- **Objective**: Determine the initial speed \( v_A \) and the height \( h \) of the ball as it passes player B.

For this problem, we will need to use the kinematic equations for projectile motion. The horizontal and vertical components of the motion are:
\[
x = v_A \cdot \cos(\theta) \cdot t
\]
\[
y = 7 + v_A \cdot \sin(\theta) \cdot t - \frac{1}{2} g t^2
\]
We can solve for \( v_A \) and \( h \) by considering the trajectory and using the distances given in the diagram.

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Would you like me to solve one or both of these problems in detail? Additionally, here are five related questions you might find useful:

1. How is the range of a projectile affected by the initial speed and launch angle?
2. What role does air resistance play in real-world projectile motion?
3. How can you calculate the time of flight for a projectile given the initial velocity and angle?
4. What is the relationship between the maximum height and the initial vertical velocity of a projectile?
5. How can you determine the optimum angle for maximum range in projectile motion?

**Tip**: When solving projectile motion problems, always break down the motion into horizontal and vertical components to apply kinematic equations separately.

1. From a video recording, a professional football player kicked a football 126 feet in 3.6 seconds. Determine the initial speed and the angle at which it was kicked. 2. A basketball shot passed over player B, barely clearing his hands. Determine the initial velocity and height of the ball when it passes over player B.

Solve real-world projectile motion problems involving a football kick and a basketball shot. Learn how to calculate initial speed, angle of launch, and the height of the projectile using kinematic equations and trigonometry. Suitable for advanced high school or college students.

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