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? - High Altitude Science
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:
📰 Discover the Perfect Light Spring Color Palette That’s Taking Over 2024! 📰 This Light Spring Color Palette Will Transform Your Home & Wardrobe! 📰 Why Every Season Should Embrace This Dazzling Light Spring Color Palette 📰 Hell Divers 2 Exposedworld Record Moments That Will Blow Your Mind 📰 Hell Is Emptydevils Gathering Unleashed You Never Saw It Coming 📰 Hell Is Emptyreal Devils Are Out Heres What Happens Next Shocking 📰 Hell Let Loose On Ps5 Overnight Viral Game Shock Youll Embrace 📰 Hell Let Loose On Ps5 The Most Unbelievable Gameplay You Must See 📰 Hell On Wheels Is Herewitness How This Chaos Transforms Cities Forever 📰 Hell On Wheels The Inferno Ride That No One Talks Aboutwatch Now 📰 Hell On Wheels Unleashed Why This Chaotic Journey Will Blow Your Mind 📰 Hell On Wheels What This Apocalyptic Ride Actually Does To Your Soul 📰 Hellblade 2 Secrets Revealed The Hidden Features That Will Shock You 📰 Hellblade 2S Game Changing Graphics Will Blow Your Mind Heres Why 📰 Hellblade Experience The Packed With Mind Bending Audio And Emotional Depth 📰 Hellblade Fans Are Shockeduncovered Sacrifice Secrets You Need To Watch Now 📰 Hellblade Gaming Secrets Why This Game Relurches The Nervous System 📰 Hellblade Senuas Sacrifice Revealed 7 Devastating Moments That Will Haunt You ForeverFinal 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.
