A projectile is launched with an initial velocity of 50 m/s at an angle of 30 degrees above the horizontal. Calculate the maximum height reached. Use \(g = 9.8 \, ext{m/s}^2\).
["### Projectile Motion: Max Height Calculated for 50 m/s at 30°", "When an object is launched into the air with speed and angle, its motion follows predictable patterns governed by physics principles. In this article, we analyze a projectile launched with an initial velocity of 50 m/s at a 30-degree angle above the horizontal, and calculate the maximum height it reaches using standard projectile motion equations.", "---", "### Understanding Projectile Motion Basics", "Projectile motion involves motion in two dimensions—horizontal and vertical—under the constant acceleration due to gravity ((g = 9.8 , ext{m/s}^2)), acting downward. While horizontal velocity remains constant (ignoring air resistance), vertical velocity changes due to gravity, causing the projectile to rise and then fall.", "---", "### Key Goal: Find Maximum Height", "The maximum height of a projectile is determined by its initial vertical velocity and gravity. At the peak of its trajectory, the vertical component of velocity becomes zero. We use kinematic equations to compute this maximum altitude.", "---", "### Step 1: Initial Vertical Velocity Component", "Given:\n- Initial speed ((v_0)) = 50 m/s\n- Launch angle (( \ heta )) = 30°\n- Acceleration due to gravity ((g)) = 9.8 m/s²", "The vertical component of the initial velocity ((v_{0y})) is:", "[\nv_{0y} = v_0 \cdot \sin(\ heta)\n]", "[\nv_{0y} = 50 \cdot \sin(30^\circ) = 50 \cdot 0.5 = 25 , ext{m/s}\n]", "---", "### Step 2: Use Kinematic Equation to Find Maximum Height", "At maximum height, vertical velocity (v_y = 0). The equation relating initial vertical velocity, acceleration, and displacement is:", "[\nv_y^2 = v_{0y}^2 - 2g h\n]", "Set (v_y = 0) to find maximum height (h):", "[\n0 = (25)^2 - 2 \cdot 9.8 \cdot h\n]", "[\n625 = 19.6h\n]", "[\nh = \frac{625}{19.6} \approx 31.89 , ext{m}\n]", "---", "### Conclusion", "When a projectile is launched with an initial speed of 50 m/s at 30° above the horizontal, and assuming no air resistance, its maximum height is approximately 31.89 meters. This calculation relies on resolving the initial velocity into vertical components and applying the equations of motion under constant gravitational acceleration.", "---", "### Key Takeaway", "Understanding projectile motion helps in applications ranging from sports to engineering. Calculating maximum height provides insight into the peak performance or range of projectiles — essential in physics and real-world planning.", "---", "Keywords: projectile motion, maximum height, kinematics, vertical velocity, gravitational acceleration, 50 m/s projectile, 30 degree launch angle, 9.8 m/s², physics calculation."]
