A projectile is launched from the ground with an initial speed of 50 m/s at an angle of 30 degrees to the horizontal. Calculate the maximum height reached by the projectile. Use \( g = 9.8 \, \text{m/s}^2 \).

["Projectile Motion: Calculating Maximum Height from Launch Speed and Angle", "When a projectile is launched from ground level, its motion follows predictable physics laws governed by gravity. A classic example involves launching at an initial speed and angle. In this article, we analyze a projectile launched at 50 m/s at a 30° angle to the horizontal, calculating the maximum height it reaches.", "---", "### Understanding Projectile Motion", "In projectile motion, the vertical component of initial velocity determines the peak altitude. Even though the projectile moves forward, its upward vertical speed instantly slows down under gravity until it momentarily stops at the apex.", "Given:\n- Initial speed ( v_0 = 50 , \ ext{m/s} )\n- Launch angle ( \ heta = 30^\circ )\n- Acceleration due to gravity ( g = 9.8 , \ ext{m/s}^2 )", "Only the vertical component of velocity contributes to height.", "---", "### Step 1: Resolve the Initial Velocity Vertically", "The vertical component of initial velocity is:", "[\nv_{y0} = v_0 \sin \ heta\n]", "Substituting values:", "[\nv_{y0} = 50 \ imes \sin(30^\circ) = 50 \ imes 0.5 = 25 , \ ext{m/s}\n]", "---", "### Step 2: Use Kinematics to Find Maximum Height", "At maximum height, the vertical velocity becomes zero (( v_y = 0 )). Using the kinematic equation:", "[\nv_y^2 = v_{y0}^2 - 2gh_{\ ext{max}}\n]", "Set ( v_y = 0 ) and solve for ( h_{\ ext{max}} ):", "[\n0 = (25)^2 - 2 \ imes 9.8 \ imes h_{\ ext{max}}\n]", "[\n625 = 19.6 , h_{\ ext{max}}\n]", "[\nh_{\ ext{max}} = \frac{625}{19.6} \approx 31.89 , \ ext{meters}\n]", "---", "### Final Result", "The projectile reaches a maximum height of approximately 31.89 meters. This calculation is fundamental in physics and engineering, aiding in understanding trajectories in sports, ballistics, and aerospace.", "---", "### Key Takeaway", "For vertical launch analysis:\n- Split velocity using ( v_{y0} = v_0 \sin\ heta )\n- Use ( v_y^2 = v_{y0}^2 - 2gh ) at peak height\n- Rearrange to solve for maximum height ( h )", "This simple yet powerful method enables precise trajectory predictions with gravity as the only force (ignoring air resistance).", "---", "Keywords: projectile motion, launch angle, vertical velocity, maximum height calculation, maximum height physics, kinematics, gravity effect, 30 degree projectile, 50 m/s initial speed, projectile height formula.", "---", "Need more projectile calculations? Try launching with different angles or speeds—each scenario offers unique insights into motion under constant acceleration."]