Final answer:
The total time necessary for the ball to come to rest is calculated based on the kinematic equations for motion under gravity, considering the time it takes for the ball to rise and fall back down. With an initial rise time of 2.5 seconds, the total time including the fall would be 5.0 seconds, presuming symmetrical rise and fall times.
Step-by-step explanation:
The necessary time to determine when the ball comes to rest can be calculated using kinematic equations that describe motion under the influence of gravity. In this scenario, the ball has an initial velocity of 38.3 m/s and it reaches a height where it comes to rest before falling a vertical distance of -33.3 m. Using equation D: t = sqrt(2h/g), which directly relates the time it takes for an object to fall from height h to the acceleration due to gravity g, we can find the time for the fall.
First, the ball rises to its peak where the velocity becomes zero. Subsequently, it starts falling. Given the ball is subject to Earth's gravitational acceleration, which is approximately 9.8 m/s2, we can plug this into the given formula to find the time taken for the ball to fall the last -33.3 meters to come to rest at the spectator’s hands.
Thus, the total time for the ball to come to rest includes the time it takes to rise to the top of its trajectory and the time it takes to fall back down. Since the ball rises for 2.5 seconds, with the assumption that it will take the same amount of time to fall the same distance back down, the total time to come to rest would be 2.5 seconds upwards plus 2.5 seconds downwards, resulting in a total of 5.0 seconds. However, remember that the actual distance fallen might be different, so you should calculate the total time using the initial fall height and the second fall height to make sure if the time to rest is other than 5.0 seconds.