A ball is thrown straight downward from the top of a tall building, with an initial speed of 30 ft/s. Suppose the ball strikes the ground with a speed of 190 ft/s. How tall is the building? Acceleration is -32 ft/s squared
V^2 = Vo^2 + 2g*h
h = (V^2-Vo^2)/2g
h = ((190)^2-(30)^2)/64 = 550 Fi.
height = -16t^2 -30t + d, where d is the height of the building
d(height)/dt = velocity = -32t - 30
-190 = -32t - 30
32t = + 160
t = 5
at t = 5 , height = 0
0 = -16(25) - 150 + d
550 = d
To find the height of the building, we can use the equations of motion for uniformly accelerated linear motion.
Step 1: Find the time it takes for the ball to reach the ground.
From the given information, we know the initial velocity (vi = 30 ft/s), the final velocity (vf = -190 ft/s), and the acceleration (a = -32 ft/s^2).
Using the equation vf = vi + at, we can rearrange to solve for time (t):
-190 ft/s = 30 ft/s + (-32 ft/s^2)t
Simplifying:
-190 ft/s - 30 ft/s = -32 ft/s^2t
-220 ft/s = -32 ft/s^2t
Dividing both sides by -32 ft/s^2:
t = -220 ft/s / -32 ft/s^2
t = 6.875 s
Step 2: Calculate the height of the building.
Using the equation h = vi * t + (1/2) * a * t^2, where h represents the height of the building:
h = 30 ft/s * 6.875 s + (1/2) * (-32 ft/s^2) * (6.875 s)^2
Simplifying:
h = 206.25 ft + (-11 ft/s^2) * 236.328125 s^2
h ≈ 206.25 ft + (-2595.609375) ft
h ≈ -2389.359375 ft
Since we cannot have a negative height, we can conclude that the building height is approximately 2389.36 ft.
To find the height of the building, we can use the equations of motion for constant acceleration. The equation we will use is:
v^2 = u^2 + 2as
Where:
- v is the final velocity (190 ft/s in this case),
- u is the initial velocity (30 ft/s in this case),
- a is the acceleration (-32 ft/s^2 in this case), and
- s is the distance or height traveled.
First, let's rearrange the equation to solve for s:
s = (v^2 - u^2) / (2a)
Now, we can substitute the values into the equation:
s = (190^2 - 30^2) / (2 * (-32))
s = 36100 - 900 / (-64)
s = 35200 / (-64)
s ≈ -550
The distance returned is negative because the ball is falling downwards. However, we are interested in the magnitude or absolute value of the height, so we don't need to consider the negative sign.
Therefore, the height of the building is approximately 550 feet.