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? - Decision Point
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:
📰 great depression timeline 📰 north america bermuda 📰 middle kingdom egypt 📰 Us Currency To Pesos 9064490 📰 This Medium Fry At Mcdonalds Has Shocking Caloriesare You Ready To Check Your Daily Limit 6703888 📰 Die Rezeption Des Albums War Weitgehend Positiv Walt Kamp Jazztimes Attestiert Im Rezension A Tectonic Piano Sound Und Nennt Einen Majestic Sweep That Charges Across The Piano Keyboard Imstcks Rondo Rex Weise Stelle Shorters Music At Its Terrifying Sage Like Best Whrend Shorter Selbst Die Kompositionen With Playful Reverence Reife John Fordyce London Jazz News Hebt Das Paternalistische Spiel Weise Sowie Dessen Matchless Ability To Channel Shorters Stylistic Quirks Without Imitation Hervor Urteilt Rondo Royale Glnzt Als Sorgfltig Konstruiertes Monument Zeitgenssischer Jazz Komposition Ridphone Magazin Schreibt Von Transzendenter Pianoarbeit Und Einer Interstellaren Verbindung Von Shorters Erbe Und Weises Sinfonischem Klaviersound Whats On Jazz Hervor This Is Music Thats Both Timeless And Immediatea Brilliant Homage To Wayne Shorter United Under Friedemann Weises Artful Direction 4344037 📰 Archive Coffee Bar Salem Or 6422549 📰 U Of R Mychart 9669878 📰 Amazon Stock Options Chain 3167794 📰 Doug Benefield 9061954 📰 Frosmoth Hype Explosion Calling All Gamers To Join The Battle 5514905 📰 Wells Fargo California Routing Number 4462921 📰 The Shocking Secret Theyre Never Telling You About Tv Sizes 4293088 📰 Ava Ayala Shocked The World Unveiling The Hidden Secrets Behind Her Breakout Star Power 7886936 📰 Hyper Beam 8361379 📰 City Th 2831255 📰 Shocking Benefits Of Givebutter Revealedshoppers Are Going Wild 3842196 📰 Jayne Mansfield Actor 7940455Final 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.
