A golf ball is hit at ground level. The ball is
observed to reach its maximum height above
ground level 2.2 s after being hit. 1.3 s after
reaching this maximum height, the ball is
observed to barely clear a fence that is 780 ft
from where it was hit.
The acceleration of gravity is 32 ft/s
2
.
How high is the fence?
To determine the height of the fence, we can use the equations of motion for an object in free-fall.
We know that the acceleration due to gravity is 32 ft/s^2, and we are given two time intervals:
1) From the time the ball is hit to the time it reaches its maximum height, which is 2.2 seconds.
2) From the time the ball reaches its maximum height to the time it barely clears the fence, which is 1.3 seconds.
First, let's find the initial velocity of the ball when it was hit:
The time taken to reach the maximum height, t1 = 2.2 seconds.
Using the formula for vertical displacement, we have:
s = ut + (1/2)at^2
At the maximum height, the ball has no vertical displacement (s=0), so we can rewrite the equation as:
0 = u * t1 + (1/2) * (-32) * (t1^2)
Substituting the values, we have:
0 = u * 2.2 - 16 * (2.2)^2
Solving for the initial velocity, u:
4.84u = 77.44
u = 77.44 / 4.84
u = 16 ft/s (approximately)
Now, let's find the height of the fence:
Since the fence is barely cleared, the vertical displacement at that point is equal to the height of the fence.
We know that the time taken from the maximum height to barely clearing the fence, t2 = 1.3 seconds.
Using the formula for vertical displacement again, we have:
s = ut + (1/2)at^2
Substituting the values, we have:
s = 16 * 1.3 + (1/2) * (-32) * (1.3)^2
Simplifying the equation:
s = 20.8 - 27.04
s = -6.24 ft
Since the displacement is negative, it means the ball is 6.24 ft below the initial height when it barely clears the fence. Therefore, the height of the fence is 6.24 ft above the ground level.