Final answer:
Using physics kinematic equations, it can be shown that a baseball thrown upwards at 100 ft/s from 7.5 ft cannot reach the 208 ft. Astrodome ceiling. To hit the ceiling, a higher initial velocity would be needed.
Step-by-step explanation:
The question involves physics principles related to projectile motion and kinematics. When addressing the problem, we'll consider gravity's effect and use the kinematic equations to solve for the maximum height reached by the baseball and the necessary initial velocity to reach the Astrodome's ceiling.
Part A: Maximum Height Calculation
Using the kinematic equation for vertical motion (v^2 = u^2 + 2as), where v is the final velocity (0 ft/s at maximum height), u is the initial velocity (100 ft/s), a is the acceleration (gravity, which is -32.2 ft/s^2), and s is the displacement, we can solve for the maximum height:
0 = (100)^2 + 2*(-32.2)*(s - 7.5)
From this, we find that the maximum height s is around 159.4 ft when we add the 7.5 ft starting height. Since this value is less than the Astrodome ceiling height of 208 ft, the pitcher in this scenario cannot hit the ceiling with a speed of 100 ft/s from a height of 7.5 ft.
Part B: Initial Velocity Required
To find the initial velocity required to reach the Astrodome's ceiling, we rearrange the same kinematic equation,
u = sqrt(v^2 - 2as)
Where s is now the ceiling height minus the release height (200.5 ft), and a and v remain unchanged. Solving for u gives us the required initial velocity to just touch the ceiling. The calculated initial velocity exceeds 100 ft/s.