Answer :
We start with the formula for the speed of an object in free fall:
[tex]$$
v = \sqrt{2gh}.
$$[/tex]
To find the height [tex]$h$[/tex], we first square both sides of the equation:
[tex]$$
v^2 = 2gh.
$$[/tex]
Now, solve for [tex]$h$[/tex] by dividing both sides by [tex]$2g$[/tex]:
[tex]$$
h = \frac{v^2}{2g}.
$$[/tex]
Given that the speed when the hammer hits the floor is [tex]$v = 8$[/tex] feet per second and the acceleration due to gravity is [tex]$g = 32$[/tex] feet per second[tex]$^2$[/tex], we substitute these values into the equation:
[tex]$$
h = \frac{8^2}{2 \cdot 32}.
$$[/tex]
Calculate the numerator:
[tex]$$
8^2 = 64,
$$[/tex]
and the denominator:
[tex]$$
2 \cdot 32 = 64.
$$[/tex]
Thus, the height is
[tex]$$
h = \frac{64}{64} = 1.0 \text{ foot}.
$$[/tex]
So, the hammer was dropped from [tex]$\boxed{1.0 \text{ foot}}$[/tex] above the ground.
[tex]$$
v = \sqrt{2gh}.
$$[/tex]
To find the height [tex]$h$[/tex], we first square both sides of the equation:
[tex]$$
v^2 = 2gh.
$$[/tex]
Now, solve for [tex]$h$[/tex] by dividing both sides by [tex]$2g$[/tex]:
[tex]$$
h = \frac{v^2}{2g}.
$$[/tex]
Given that the speed when the hammer hits the floor is [tex]$v = 8$[/tex] feet per second and the acceleration due to gravity is [tex]$g = 32$[/tex] feet per second[tex]$^2$[/tex], we substitute these values into the equation:
[tex]$$
h = \frac{8^2}{2 \cdot 32}.
$$[/tex]
Calculate the numerator:
[tex]$$
8^2 = 64,
$$[/tex]
and the denominator:
[tex]$$
2 \cdot 32 = 64.
$$[/tex]
Thus, the height is
[tex]$$
h = \frac{64}{64} = 1.0 \text{ foot}.
$$[/tex]
So, the hammer was dropped from [tex]$\boxed{1.0 \text{ foot}}$[/tex] above the ground.
