Understanding the Physics
* Projectile Motion: The sphere's motion is governed by projectile motion, which is influenced by gravity and the initial launch conditions.
* Vertical Component: The vertical component of the sphere's velocity determines its height.
* Gravity: Gravity constantly acts downwards, slowing the sphere's upward motion and eventually causing it to fall back down.
Formula
The maximum height (H) of a projectile can be calculated using the following formula:
H = (v₀² * sin²θ) / (2 * g)
Where:
* v₀: Initial velocity of the sphere (m/s)
* θ: Launch angle (the angle the cannon is pointed relative to the horizontal)
* g: Acceleration due to gravity (approximately 9.8 m/s²)
Explanation
* Initial Velocity (v₀): A higher initial velocity will result in a greater maximum height.
* Launch Angle (θ): The maximum height is achieved when the launch angle is 90 degrees (straight up). At other angles, the sphere has a horizontal component of velocity, reducing its vertical reach.
* Gravity (g): Gravity plays a crucial role in determining how quickly the sphere slows down and falls back down.
Example
Let's say a cannon fires a sphere with an initial velocity of 50 m/s at an angle of 45 degrees.
1. Calculate sin²θ: sin(45°) = 0.707, so sin²(45°) = 0.5
2. Plug values into the formula: H = (50² * 0.5) / (2 * 9.8)
3. Calculate: H ≈ 63.8 m
Important Notes
* Air Resistance: This formula doesn't account for air resistance, which will decrease the actual maximum height.
* Real-World Factors: Other factors like wind, the mass of the sphere, and the cannon's design can also influence the result.
Let me know if you'd like to explore how to calculate the range of the projectile (how far it travels horizontally) or if you have specific values for the variables and want to calculate the maximum height for a specific scenario!