Select the correct answer. Tara and Mallory are practicing their passing at volleyball practice. Tara bumps the ball from a height of 3 feet above the ground with an initial up velocity of 12 feet/second. At the same time, Mallory serves the ball from a height of 3.25 feet above the ground with an initial upward velocity of 9 feet/second. If \( x \) represents the number of seconds since each ball is passed and \( y \) represents the height of the ball, which system of equations models the pr motion of the two balls? Recall that the acceleration due to gravity is \( \mathbf{- 3 2} \) feet/second. A. \( y=-16 x^{2}+12 x+3 \) \[ y=-16 x^{2}+9 x+3.25 \] B. \( y=-16 x^{2}+9 x-3 \) \( y=-16 x^{2}+12 x-3.25 \) c. \( y=-16 x^{2}+12 x-3 \) \( y=-16 x^{2}+9 x-3.25 \) D. \( y=-16 x^{2}+9 x+3 \) \( y=-16 x^{2}+12 x+3.25 \)

The correct answer is A. To model the motion of the balls, we use the equations for projectile motion under gravity. For Tara's ball, starting at 3 feet with an initial upward velocity of 12 feet/second, the equation is \( y = -16x^2 + 12x + 3 \). For Mallory's ball, starting at 3.25 feet with an initial upward velocity of 9 feet/second, the equation is \( y = -16x^2 + 9x + 3.25 \). Thus, option A accurately represents the motion of both balls! Understanding how these equations are derived can greatly enhance your appreciation for physics. Both balls are subject to gravity, which acts as a constant acceleration downward, represented by the -16 in front of the \( x^2 \) term (since it’s measured in feet per second squared). So, every second, their velocities decrease because of gravity, leading to a parabolic trajectory! If you're keen on diving deeper, consider exploring the fascinating world of projectile motion through textbooks or online courses focused on physics. Concepts like initial velocity, height, and gravity make for exciting real-world applications, from sports to engineering! Websites like Khan Academy or YouTube offer great visual explanations that can clarify these concepts and make learning fun!