A ball is thrown vertically upward from the top of a building 192 feet tall with an initial velocity of 64 feet per second. The distance \( s \) (in feet) of the ball from the ground after t seconds is \( s(t)=192+64 t-16 t^{2} \). (a) After how many seconds does the ball strike the ground? (b) After how many seconds will the ball pass the top of the building on its way down? (a) After seconds the ball strikes the ground. (b)

To find when the ball strikes the ground, we need to set \( s(t) = 0 \) and solve the equation: \[ 0 = 192 + 64t - 16t^2 \] Rearranging gives us: \[ 16t^2 - 64t - 192 = 0 \] Dividing through by 16: \[ t^2 - 4t - 12 = 0 \] Now, applying the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 1, b = -4, c = -12 \): \[ t = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \] This simplifies to: \[ t = \frac{4 \pm \sqrt{16 + 48}}{2} \] \[ t = \frac{4 \pm \sqrt{64}}{2} \] \[ t = \frac{4 \pm 8}{2} \] This gives us two solutions: 1. \( t = 6 \) seconds (when it hits the ground) 2. \( t = -2 \) seconds (discarded as time can't be negative) For part (b), to find when the ball passes the top of the building again on its way down, set \( s(t) = 192 \): \[ 192 = 192 + 64t - 16t^2 \] This simplifies to: \[ 0 = 64t - 16t^2 \] \[ 0 = 16t(4 - t) \] This gives us \( t = 0 \) (the initial throw) and \( t = 4 \) seconds (when it descends back past the same height). So the answers are: (a) 6 seconds, (b) 4 seconds. --- The concept of projectile motion dates back to Aristotle and Galileo, who studied the influences of gravity and initial velocities long before formulas like these were established. In fact, Galileo is credited with laying the groundwork for kinematics, which is vital in understanding objects moving in two dimensions! You can see projectile motion at work in real life every time you toss a basketball towards a hoop! The path it takes is a parabola, similar to the motion we discussed with the ball from the building. Understanding these principles can help in sports, engineering, and even video game physics to create more accurate simulations of motion!