A ball is thrown upward with an initial velocity of 49 m/s. Using the formula \(h(t) = -4.9t^2 + 49t\), what is the maximum height reached? - Deep Underground Poetry
Maximum Height of a Ball Thrown Upward: A Physics Perspective Using the Quadratic Formula
Maximum Height of a Ball Thrown Upward: A Physics Perspective Using the Quadratic Formula
When a ball is thrown upward with an initial velocity of 49 meters per second, its height over time follows a quadratic trajectory governed by the equation:
\[
h(t) = -4.9t^2 + 49t
\]
Understanding the Context
This formula models the vertical position \(h(t)\) in meters as a function of time \(t\) in seconds. Because the coefficient of \(t^2\) is negative, the parabola opens downward, meaning the ball eventually rises to a peak and then falls back down — a classic example of projectile motion under gravity (with air resistance neglected).
Finding the Maximum Height
The maximum height occurs at the vertex of the parabola. For any quadratic function in the form \(h(t) = at^2 + bt + c\), the time \(t\) at which the peak height is reached is given by:
\[
t_{\ ext{max}} = -\frac{b}{2a}
\]
Image Gallery
Key Insights
In our specific formula, \(a = -4.9\) and \(b = 49\). Substituting these values:
\[
t_{\ ext{max}} = -\frac{49}{2 \ imes (-4.9)} = \frac{49}{9.8} = 5 \ ext{ seconds}
\]
Now substitute \(t = 5\) into the height equation to find the maximum height:
\[
h(5) = -4.9(5)^2 + 49(5) = -4.9(25) + 245 = -122.5 + 245 = 122.5 \ ext{ meters}
\]
Conclusion
🔗 Related Articles You Might Like:
📰 Wells Fargo Bonus Checking 📰 Private Personal Loan 📰 Wells Fargo Wallet 📰 Glyph Shocking Encounter With A 78 Year Old Mentor Who Transformed My Perspective 6776359 📰 Nota Alma Rebellion Fue Lanzado En 2022 Y Combina Elementos Explosivos De Documental Con Un Tono Cercano Y Motivador Ideal Para Plataformas Digitales Enfocadas En Realidad Joven Y Creatividad 3189434 📰 Vertical Extent 15 Cm Half Of 3 Cm 3670864 📰 Kim Paisley 1778531 📰 Block Champ Game Files This Simple Game Dominates All Competitive Leaderboards 3477085 📰 Alan Wake Remaster Pc 1398769 📰 Finally Learn How To Install Solver In Excel Without Struggling 2324547 📰 Queens University 4517067 📰 Dont Miss Thisfidelity Atp Online Now Exclusive Download Link Bonus 2970538 📰 New South Park Episodes 7211788 📰 Are Nacho Fries Back 7951427 📰 The Shocking Truth What Wild Rabbits Eat To Survive In Nature 8118255 📰 Harry Potter Streaming 622719 📰 Free Goodies Flood Inyour Birthday Comes Packed With Secret Truly Worth 5866848 📰 Best Buy Xbox Series X The Absolute Best Deal For Gamers In 2024 862723Final Thoughts
The maximum height reached by the ball is 122.5 meters. This result illustrates how algebraic expressions rooted in motion physics can precisely predict real-world phenomena, helping students and scientists alike understand the beauty of parabolic motion and optimal performance in projectile dynamics.
Keywords: ball thrown upward, maximum height formula, quadratic motion physics, projectile motion, h(t) = -4.9t² + 49t, vertex of a parabola, maximizing height, physics education.
