https://file.aiquickdraw.com/mathbot/file/1727325382057z3glmgoo.jpg

Let's break down the two problems in the image:

### Problem 1: Football Kick
- A football is kicked 126 feet, and it takes 3.6 seconds to travel this distance.
- You are asked to determine:
  - The initial speed \(v_0\)
  - The launch angle \(\theta\) at which the football was kicked.

This is a projectile motion problem. The following key equations apply to projectile motion:
1. The horizontal motion: \( x = v_0 \cdot \cos(\theta) \cdot t \)
2. The vertical motion: \( y = v_0 \cdot \sin(\theta) \cdot t - \frac{1}{2} g t^2 \)

Since the ball travels a horizontal distance of 126 ft in 3.6 seconds, we can use the horizontal equation to solve for \(v_0\cos(\theta)\). Given no vertical displacement (starting and ending at the same height), we can also use symmetry to help deduce \(v_0\sin(\theta)\). Solving these equations together will give us both the initial speed and the angle.

### Problem 2: Basketball Shot
- The basketball is shot at an angle of 30° with respect to the horizontal.
- The ball just clears player \(B\), who is standing 25 feet away from the point of the shot, and reaches a height \(h\).
- You need to find:
  - The magnitude of the initial velocity \(v_A\)
  - The height \(h\) of the ball when it passes over player \(B\).

This is another projectile motion problem, where you need to apply the following equations:
1. The horizontal displacement: \(x = v_A \cdot \cos(30^\circ) \cdot t\)
2. The vertical displacement: \(y = v_A \cdot \sin(30^\circ) \cdot t - \frac{1}{2} g t^2\)

You know that the ball travels 25 feet horizontally before reaching player \(B\) and that the vertical displacement is \(h\) at that point.

I can now proceed with solving these problems. Would you like me to solve them step by step?

From a video recording, it was observed that a professional football player kicked a football 126 feet during a measured time of 3.6 seconds. Determine the initial speed of the ball and the angle θ at which it was kicked.
Measurements of a shot recorded during a basketball game are shown. The ball passed through the hoop even though it barely cleared the hands of player B who attempted to block it. Determine the magnitude v_A of its initial velocity and the height h of the ball when it passes over player B.

Solve two projectile motion problems involving a football kick and a basketball shot. Learn how to calculate the initial velocity, launch angle, and height of the ball using kinematic equations and trigonometry. Suitable for college-level physics.

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