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? - Simpleprint
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 Secrets Behind Esto Memorable Mandala Tattoo Designs! 📰 Shocked by This Stunning Mandala Tattoo? Here’s Why Artists Are Obsessed! 📰 ento unique mandala tattoo styles you’ll Want to转发—and Insta-famous! 📰 3 Playstation Refund Hassle This Secret Method Works Fastclick To See How 📰 3 Playstation Wrapped 2024 Valued Over 1000 Heres What They Dropped 📰 3 Pokeball Plus Revealed This Feature Will Change How You Catch Pokmon Forever 📰 3 Pokemon Games On Switch These 5 Are The Most Played No Regrets 📰 3 Pokmon 1St Movie The Secrets Hidden In Every Frame You Need To Watch Now 📰 3 Pokmon Go Wild Spawns Are Explodingtop Spawn Zones You Need To Check Now 📰 3 Pokmon Go Wild Spawns Are Hitting New Peakssee Which Areas Are Lit Up 📰 3 Pokmon Legends Mega Evolutions Breakthroughwhat Players Are Obsessed About 📰 3 Pokmon Movie I Choose Youll Watch It Twice Because This Ending Is Unstoppable 📰 3 Pokmon Renegade Platinum Secrets You Need To Unlock Before The End Game 📰 3 Pokmon Sun Revealed The Phenomenal Strategies Every Trainer Must Master Today 📰 3 Pokmon Team Rocket Shocked Fans Exclusive Breakdown Of Their Most Dangerous Rivalry 📰 3 Prokerala Breakdown 7 Unmissable Truths About Keralas Magic 📰 3 Ps4 Pro Vs Five Years Ago The Breakthrough Upgrade Thatll Blow Your Console Game 📰 3 Secret Clean Gaming Hack For The Best Playground Game Experience Try It NowFinal 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.
