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Answer :
To solve this problem, we need to find out how high above the ground the hammer was when it was dropped. We are given the formula for the velocity of an object in free fall:
[tex]\[ v = \sqrt{2gh} \][/tex]
Where:
- [tex]\( v \)[/tex] is the final velocity of the object (12 feet per second in this case).
- [tex]\( g \)[/tex] is the acceleration due to gravity (32 feet/second²).
- [tex]\( h \)[/tex] is the height from which the object was dropped.
We want to solve for [tex]\( h \)[/tex]. Let's break it down step by step:
1. Rearrange the formula to solve for [tex]\( h \)[/tex]:
[tex]\[ v^2 = 2gh \][/tex]
Dividing both sides by [tex]\( 2g \)[/tex]:
[tex]\[ h = \frac{v^2}{2g} \][/tex]
2. Plug in the given values:
- [tex]\( v = 12 \)[/tex] feet/second
- [tex]\( g = 32 \)[/tex] feet/second²
Substituting these values into the equation:
[tex]\[ h = \frac{(12)^2}{2 \times 32} \][/tex]
3. Calculate the result:
- First, calculate [tex]\( 12^2 \)[/tex], which is 144.
- Then calculate [tex]\( 2 \times 32 \)[/tex], which is 64.
Now divide 144 by 64:
[tex]\[ h = \frac{144}{64} = 2.25 \][/tex]
Therefore, the height from which the hammer was dropped is 2.25 feet. So, the correct answer is option D: 2.25 feet.
[tex]\[ v = \sqrt{2gh} \][/tex]
Where:
- [tex]\( v \)[/tex] is the final velocity of the object (12 feet per second in this case).
- [tex]\( g \)[/tex] is the acceleration due to gravity (32 feet/second²).
- [tex]\( h \)[/tex] is the height from which the object was dropped.
We want to solve for [tex]\( h \)[/tex]. Let's break it down step by step:
1. Rearrange the formula to solve for [tex]\( h \)[/tex]:
[tex]\[ v^2 = 2gh \][/tex]
Dividing both sides by [tex]\( 2g \)[/tex]:
[tex]\[ h = \frac{v^2}{2g} \][/tex]
2. Plug in the given values:
- [tex]\( v = 12 \)[/tex] feet/second
- [tex]\( g = 32 \)[/tex] feet/second²
Substituting these values into the equation:
[tex]\[ h = \frac{(12)^2}{2 \times 32} \][/tex]
3. Calculate the result:
- First, calculate [tex]\( 12^2 \)[/tex], which is 144.
- Then calculate [tex]\( 2 \times 32 \)[/tex], which is 64.
Now divide 144 by 64:
[tex]\[ h = \frac{144}{64} = 2.25 \][/tex]
Therefore, the height from which the hammer was dropped is 2.25 feet. So, the correct answer is option D: 2.25 feet.
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