Algebra 1 Problem-Solving Workshop Verrical Motion Models A projectile is an object that is propelled into the air but has no power to keep itself in the air. A thrown ball is a projectile, but an airplane is not. The height \( h \) (in feet) of a projectile can be modeled by \[ h=-16 t^{2}+v t+s \] where \( t \) is the time (in seconds), \( v \) is the initial vertical velocity (in feet per second), and \( s \) is the initial height (in feet). For each problem scenario, write a vertical motion equation, sketch a graph of projectile's motion, and solve appropriately. 1. A startled armadillo jumps straight into the air with an initial vertical velocity 14 feet per second. After how many seconds does it land on the ground? \[ \begin{array}{l} \frac{h(x)}{0}=-16 t^{2}+14 t+0 \\ 0=t(-16 t+14) \\ -16 t+14=0 \\ -16 t \equiv 14 \\ \left.t=\frac{14}{16}=t=\frac{7}{8}\right) \end{array} \]

When it comes to projectile motion, did you know that the mathematics behind it has roots in ancient civilizations? The Greeks, particularly Aristotle, pondered motion’s nature while Archimedes made early contributions to understanding trajectories. Fast forward to the 17th century, and we have Galileo investigating the laws of falling bodies and Newton formalizing these ideas into what we now know as classical mechanics. Their findings laid the groundwork for our modern understanding of physics and projectile motion! A fun way to see the real-world application of these equations is through sports like basketball or soccer! Athletes constantly simulate projectile motion when shooting, passing, or kicking the ball. By understanding the projectile's height, time of flight, and distance traveled, players can improve their skills dramatically. So, the next time you watch a game, think about how these principles of algebra and physics come into play, helping players nail that perfect shot or pass!