5. What relationship is observed between the launch angle and the maximum height of the projectile?
Answer (1)
Answer:The relationship between the launch angle and the maximum height of a projectile is **sinusoidal**. Here's why:* **Maximum Height Formula:** The maximum height (H) of a projectile is determined by the following equation: H = (v₀² * sin²θ) / (2g) where: * v₀ is the initial velocity * θ is the launch angle * g is the acceleration due to gravity* **Sinusoidal Relationship:** The formula shows that the maximum height is directly proportional to the square of the sine of the launch angle (sin²θ). The sine function produces a sinusoidal curve, meaning it oscillates between a maximum and minimum value.* **Key Points:** * **Maximum at 90°:** The maximum height is achieved when the launch angle is 90 degrees (straight up), as sin(90°) = 1. * **Zero at 0° and 180°:** The maximum height is zero when the launch angle is 0 degrees (horizontal) or 180 degrees (horizontal in the opposite direction), as sin(0°) = sin(180°) = 0. * **Symmetry:** The maximum height is the same for launch angles θ and (180° - θ). This means that a projectile launched at 30° will reach the same maximum height as one launched at 150°.**In Summary:** As the launch angle increases from 0° to 90°, the maximum height of the projectile increases, reaching its peak at 90°. Then, as the launch angle increases further from 90° to 180°, the maximum height decreases back to zero. This relationship is described by the sinusoidal behavior of the sine function within the maximum height formula.
