A soccer ball rolls out of a gym through the center of a doorway into the next room. The adjacent room is \(6.00 \mathrm{~m}\) by \(6.00 \mathrm{~m}\) with the \(2.00-\mathrm{m}\) wide doorway located at the center of the wall. The ball hits the center of a side wall at \(45.0^{\circ} .\) If the coefficient of restitution for the soccer ball is \(0.700,\) does the ball bounce back out of the room? (Note that the ball rolls without slipping, so no energy is lost to the floor.)

First, let's find the time it takes for the ball to travel from the doorway to the center of the room, which is half of the room's width, i.e., 3 meters. Then, we will calculate the initial speed. Let's use the formula: d = (1/2)at^2 Considering d = 3m and a = g*cos(45), where g = 9.81m/s^2 (acceleration due to gravity), we can find the time, t. 3 = (1/2)(9.81*cos(45))t^2 By solving this equation, we get: t = 0.7744 s Now, let's find the initial speed (v) of the ball using the formula: v = at v = (9.81*cos(45))*0.7744 v = 5.38 m/s

Now, we need to find the final speed of the ball after bouncing off the wall. The coefficient of restitution (e) is given as 0.700, and the relation for the final speed (vf) is: vf = ev vf = 0.700*(5.38) vf = 3.77 m/s

Since the ball hits the wall at a 45-degree angle and maintains this angle during the bounce, the angle at which the ball bounces off the wall will also be \(45.0^{\circ}\).

Now, we need to determine the distance traveled by the ball after bouncing off the wall and the time it takes to do so. Let's use the following expression to find the time (t_bounce) taken after the bounce: t_bounce = 2*(vf*sin(45))/g t_bounce = 2*(3.77*sin(45))/9.81 t_bounce = 0.5366 s Next, we need to find the horizontal distance (d_bounce) traveled by the ball in this time: d_bounce = vf*cos(45)*t_bounce d_bounce = 3.77*cos(45)*0.5366 d_bounce = 1.81 m Since the doorway is 2 meters wide and located in the center of the 6-meter wall, the room's center lies 1 meter away from the edge of the doorway. Therefore, since the ball travels a horizontal distance of 1.81 meters after bouncing, it does not reach the doorway and doesn't bounce back out of the room.