Understanding the Problem
* Range: The horizontal distance the shell travels before hitting the ground.
* Initial Velocity: The speed at which the shell is launched (1600 m/s).
* Launch Angle: The angle above the horizontal at which the shell is launched (64 degrees).
Key Concepts
* Projectile Motion: The motion of an object launched into the air, influenced by gravity.
* Horizontal Velocity: The horizontal component of the initial velocity, which remains constant throughout the flight.
* Vertical Velocity: The vertical component of the initial velocity, which is affected by gravity.
Calculations
1. Resolve the initial velocity into horizontal and vertical components:
* Horizontal component (Vx): Vx = V * cos(theta) = 1600 m/s * cos(64°) ≈ 694.3 m/s
* Vertical component (Vy): Vy = V * sin(theta) = 1600 m/s * sin(64°) ≈ 1437.5 m/s
2. Find the time of flight:
* The time it takes for the shell to reach its highest point and then fall back to the ground.
* Use the vertical component of velocity (Vy) and the acceleration due to gravity (g = 9.8 m/s²).
* Time to reach the highest point (t1): Vy = g * t1 => t1 = Vy / g ≈ 146.6 s
* Total time of flight (t): t = 2 * t1 ≈ 293.2 s
3. Calculate the range (R):
* Range is the horizontal distance covered in the time of flight.
* R = Vx * t ≈ 694.3 m/s * 293.2 s ≈ 203,560 meters
Therefore, the shell's range is approximately 203,560 meters (or 203.56 kilometers).
Important Note: This calculation assumes that air resistance is negligible. In reality, air resistance would significantly affect the shell's trajectory and range.
