A ball is thrown upward from a height of 10 meters with an initial velocity of 20 m/s. Its height is modeled by \( h(t) = -5t^2 + 20t + 10 \). When does it reach maximum height?
["Understanding the Motion of a Ball Thrown Upward: When Does It Reach Maximum Height?", "When a ball is thrown upward from an initial height of 10 meters with an initial velocity of 20 meters per second, its height above the ground is modeled mathematically by the quadratic equation:", "[\nh(t) = -5t^2 + 20t + 10\n]", "This equation describes the ball’s height ( h(t) ) in meters at time ( t ) seconds after being thrown. But when exactly does the ball reach its highest point? The answer lies in recognizing that this is a parabolic trajectory under constant gravity, and the vertex of the parabola represents the peak height.", "---", "### The Physics Behind Maximum Height", "In projectile motion, the vertical position follows a quadratic function due to constant downward acceleration (gravity). The general form is:", "[\nh(t) = at^2 + bt + c\n]", "where:\n- ( a = -5 ) m/s² (due to gravity, negative because it opposes upward motion),\n- ( b = 20 ) m/s (initial upward velocity),\n- ( c = 10 ) m (initial height).", "The time ( t ) at which the maximum height occurs is found using the vertex formula for a parabola:", "[\nt = -\frac{b}{2a}\n]", "Substituting the values:", "[\nt = -\frac{20}{2(-5)} = -\frac{20}{-10} = 2 \ ext{ seconds}\n]", "Thus, the ball reaches its maximum height at ( t = 2 ) seconds.", "---", "### Calculating the Maximum Height", "To find the actual height at this moment, plug ( t = 2 ) into ( h(t) ):", "[\nh(2) = -5(2)^2 + 20(2) + 10 = -5(4) + 40 + 10 = -20 + 40 + 10 = 30 \ ext{ meters}\n]", "So, the ball soars to a maximum height of 30 meters after 2 seconds.", "---", "### Why This Matters in Real-World Applications", "Understanding when the ball reaches its apex is essential in sports, engineering, and physics. High jumpers, rocket scientists, and game designers rely on such models to predict motion and optimize performance or safety. This simple quadratic model captures the essence of how gravity affects upward motion, showing that initial velocity and height together determine the highest point reached.", "---", "### Conclusion", "For a ball thrown upward from 10 meters with an initial speed of 20 m/s, modeled by ( h(t) = -5t^2 + 20t + 10 ), the peak height is reached at 2 seconds. This moment marks not just the highest point, but a key insight into the forces of motion—proving that even in simple physics, elegant mathematics reveals powerful truths.", "If you're exploring physics, projectile motion remains one of the clearest examples of quadratic relationships in real life—perfect for students, educators, and anyone curious about how direction and speed shape the skies."]
