Time for the Ball to Return to its Original Height: A Mathematical Analysis

Understanding the motion of an object, such as a ball, under the influence of gravity is a fundamental concept in physics and mathematics. In this article, we will explore how to determine the time it takes for a ball to return to its original height after being thrown upward. The height of the ball over time is modeled by the quadratic equation ht -12t2 34t 28.

Setting Up the Equation

Given the height function ht -12t2 34t 28, we need to find the time t at which the ball returns to its original height of 28 units. Therefore, we need to solve the equation -12t2 34t 28 28.

Solving the Quadratic Equation

Eliminating the Constant Term

By subtracting 28 from both sides of the equation, we simplify to -12t2 34t 0. This can be factored as follows:

[-12t2 34t 0
-2t(6t - 17) 0]

Finding the Roots

The equation is now a product of two factors set to zero. Setting each factor to zero gives us:

[ -2t 0 Rightarrow t 0 ]

and

[ 6t - 17 0 Rightarrow t frac{17}{6}]

The solution t 0 represents the moment the ball is launched, and the solution t frac{17}{6} represents the time when the ball returns to its original height. Converting this fraction to a decimal, we get t approx 2.83 seconds.

Conclusion

In conclusion, the time it takes for the ball to return to its original height after being thrown upward is frac{17}{6} seconds or approximately 2.83 seconds. This analysis demonstrates the application of quadratic equations in physics and provides insight into the behavior of projectile motion under the influence of gravity.

Understanding the underlying mathematical principles can help us better predict and analyze the motion of objects in the real world, whether it be in sports, construction, or any other field involving physical motion.