A baseball is hit at ground level. The ball is observed to reach its maximum height above ground level 3.0 s after being hit. And 2.5 s after reaching this maximum height, the ball is observed to barely clear a fence that is 97.5 m from where it was hit. How high is the fence?
A) 8.2 m
B) 15.8 m
C) 13.5 m
D) 11.0 m
E) 4.9 m

Question

Quizplus · Accepted Answer

Firstly, we can use the kinematic equations to determine the maximum height the ball reaches: 
y = v0t + (1/2)at^2 
At its maximum height, the ball is not moving vertically (v = 0) and acceleration is due to gravity (a = -9.81 m/s^2), so: 
0 = v0 + (-9.81)(3) 
v0 = 29.43 m/s

Plugging this initial velocity back into our equation, we can solve for the maximum height: 
y = (29.43)(3) + (1/2)(-9.81)(3)^2 
y = 44.145 m

Next, we can use the kinematic equations to determine the velocity of the ball at the point it clears the fence: 
y = v0t + (1/2)at^2 
97.5 = 29.43(2.5) + (1/2)(-9.81)(2.5)^2 
97.5 = 73.575 - 30.9375 
97.5 = 42.6375 + v0(2.5) 
v0 = 20.924 m/s

At this point, the ball is just high enough to clear the fence. We can use the kinematic equations once more to determine the height of the ball at this point: 
y = v0t + (1/2)at^2 
y = 20.924(2.5) + (1/2)(-9.81)(2.5)^2 
y = 52.31 m

Therefore, the height of the fence is the difference between the maximum height achieved by the ball and the height of the ball when it clears the fence: 
fence height = y(max) - y(clear) 
fence height = 44.145 - 52.31 
fence height ≈ 8.165 ≈ 8.2 m

The best choice is therefore (C), 13.5 m.

A Baseball Is Hit at Ground Level