A projectile is launched at an initial velocity of 50 m/s at an angle of 30° above the horizontal. Calculate the horizontal range of the projectile (assuming no air resistance and using \(g = 9.8 \, \text{m/s}^2\)).

["### Projectile Motion: Calculating Horizontal Range for a 50 m/s Launch at 30°", "When a projectile is launched into the air, its motion can be broken down into horizontal and vertical components. Understanding key parameters like initial velocity, launch angle, and gravitational acceleration is essential for predicting its trajectory and final landing point.", "---", "#### Key Parameters in This Scenario", "- Initial velocity (v₀): 50 m/s\n- Launch angle (θ): 30° above the horizontal\n- Gravitational acceleration (g): 9.8 m/s² (acting downward)\n- Air resistance: Neglected (ideal conditions)", "---", "#### Step 1: Resolve Initial Velocity into Components", "The horizontal and vertical components of the initial velocity determine the motion’s path.", "- Horizontal component (v₀ₓ):\n ( v_{0x} = v_0 \cdot \cos(\ heta) = 50 \cdot \cos(30^\circ) )\n Using ( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 ),\n ( v_{0x} = 50 \cdot 0.866 = 43.3 , \ ext{m/s} )", "- Vertical component (v₀ᵧ):\n ( v_{0y} = v_0 \cdot \sin(\ heta) = 50 \cdot \sin(30^\circ) )\n Since ( \sin(30^\circ) = 0.5 ),\n ( v_{0y} = 50 \cdot 0.5 = 25 , \ ext{m/s} )", "---", "#### Step 2: Use the Range Formula for Projectile Motion", "For a projectile launched from ground level and landing at the same elevation, the horizontal range (R) is given by:", "[\nR = \frac{v_0^2 \cdot \sin(2\ heta)}{g}\n]", "This formula uses the full initial velocity magnitude and accounts for the sinusoidal nature of the vertical motion under constant gravity.", "---", "#### Step 3: Plug in the Values", "[\nR = \frac{(50)^2 \cdot \sin(60^\circ)}{9.8}\n]", "Note: ( \ heta = 30^\circ \Rightarrow 2\ heta = 60^\circ ), and ( \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 )", "[\nR = \frac{2500 \cdot 0.866}{9.8} = \frac{2165}{9.8} \approx 220.92 , \ ext{meters}\n]", "---", "#### Final Answer", "The horizontal range of the projectile is approximately 220.9 meters.", "This result shows how the launch angle significantly influences range—even though 30° is not the maximum range angle (which is 45°), reaching it ensures a record-setting horizontal distance under ideal conditions.", "---", "#### Why This Matters", "Projectile motion principles underpin numerous real-world applications—from sports and ballistics to engineering and space launches. Grasping how initial velocity and angle shape the trajectory helps optimize performance and predict outcomes in physics-based systems.", "---", "Keywords: projectile motion, horizontal range, projectile launched at 50 m/s, 30 degree angle, kinematics, physics formula, gravity 9.8 m/s², sine formula range, no air resistance.", "---", "By applying fundamental trigonometric and kinematic equations, this example demonstrates the power of physics in predicting motion—elegant, accurate, and essential in engineering and science."]