Answer:
The depth of the well is 382.17 feet.
Explanation:
Here, we are to solve for the depth (y) of the well using two concepts; free fall and uniform motion. These two concepts were added to have a total time of 5 seconds which is the time for the stone to hit the water below the well (free fall) and the time to hear the splash of the water (uniform motion).
So, we have an equation:
[tex]t_{stone}+t_{sound}=[/tex] 5 equation 1
Let us now use the formula in free fall to solve for the time for the stone to hit the water, we have:
For [tex]t_{stone}[/tex]:
y = ¹/₂ gt²
t = [tex]\sqrt{\frac{2y}{g} }[/tex] equation 2
and for the time to hear the splash of the water using uniform motion, we have:
v = d/t
v = y/t
Therefore:
t = y/v equation 3
Solving the problem
Let us substitute equations 2 and 3 to equation 1, we have:
[tex]t_{stone}+t_{sound}=[/tex] 5
[tex]\sqrt{\frac{2y}{g} }[/tex] + y/v = 5
Now, let us solve for y (depth of the well) by simplifying the equation.
[tex]\frac{2y}{g} =(5-\frac{y}{v} )^2[/tex]
[tex]\frac{2y}{g} =25-\frac{10y}{v} +\frac{y^2}{v^2}[/tex]
substitute the values of v and g = 32.2 ft/s²
[tex]\frac{2y}{32.2} =25-\frac{10y}{1129} +\frac{y^2}{1129^2}[/tex]
[tex]\frac{y}{16.1} =25-\frac{y}{112.9} +\frac{y^2}{1274641}[/tex]
Simplify the equation to standard form, we have:
[tex]\frac{y^2}{1274641}-\frac{y}{16.1}-\frac{y}{112.9} +25=0[/tex]
[tex]\frac{y^2}{1274641}-\frac{129y}{1817.69} +25=0[/tex]
Then solve for the depth of the well, y using quadratic equation, we have:
y = 382.17 feet and
y = 90,072.8 feet
Let do some checking:
t = [tex]\sqrt{\frac{2y}{g} }[/tex]
t = [tex]\sqrt{\frac{2(90,072.8)}{32.2} }[/tex] = 74.79 seconds > 5 seconds which is invalid!
t = [tex]\sqrt{\frac{2(382.17)}{32.2} }[/tex] = 4.87 seconds < 5 seconds which is valid.
Therefore, the depth (y) of the well is 382.17 feet.
To learn more, just click the following links:
- Recommendations for a free falling bodies
https://brainly.ph/question/2162209
- Vertical component of a motion
https://brainly.ph/question/2604535
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