Understanding the Problem

* Initial Velocity: The ball starts with a speed of 30 m/s at an angle of 30 degrees above the horizontal.

* Horizontal and Vertical Components: We need to break the initial velocity into its horizontal (Vx) and vertical (Vy) components.

* Gravity: The only force acting on the ball after it's launched is gravity, which causes a downward acceleration of approximately 9.8 m/s².

Calculations

1. Horizontal and Vertical Components of Initial Velocity

* Vx = V * cos(theta) = 30 m/s * cos(30°) = 25.98 m/s

* Vy = V * sin(theta) = 30 m/s * sin(30°) = 15 m/s

2. Time in the Air (Time of Flight)

* Understanding: The ball goes up, reaches its highest point, and then falls back down. The time it takes to go up is the same as the time it takes to fall down.

* Vertical Motion: We'll use the vertical component of velocity (Vy) and gravity to find the time it takes to reach the highest point.

* Equation: Vy = g * t (where g is the acceleration due to gravity, and t is the time to reach the highest point)

* Solving for t: t = Vy / g = 15 m/s / 9.8 m/s² = 1.53 s

* Total Time in Air: The total time in the air is twice the time to reach the highest point: 1.53 s * 2 = 3.06 s

3. Horizontal Distance (Range)

* Understanding: The horizontal distance traveled depends on the horizontal velocity and the time in the air.

* Equation: Range (R) = Vx * Time

* Solving for R: R = 25.98 m/s * 3.06 s = 79.64 m

4. Maximum Height

* Understanding: The maximum height occurs when the vertical velocity becomes zero (at the peak of the trajectory).

* Equation: Vy² = Uy² + 2 * g * h (where Uy is the initial vertical velocity, and h is the maximum height)

* Solving for h: 0 = 15² + 2 * (-9.8) * h

* h = 15² / (2 * 9.8) = 11.48 m

Summary

* Time in the air (Time of flight): 3.06 seconds

* Horizontal distance (Range): 79.64 meters

* Maximum height: 11.48 meters