Answer :
We start with the equation that relates the speed of an object under free fall to the height from which it was dropped:
[tex]$$
v = \sqrt{2gh}
$$[/tex]
where
- [tex]$v$[/tex] is the final speed (in feet per second),
- [tex]$g$[/tex] is the acceleration due to gravity (in feet per second squared), and
- [tex]$h$[/tex] is the height above the ground (in feet).
Since the hammer hits the floor at [tex]$12 \text{ ft/s}$[/tex], we have:
[tex]$$
v = 12 \text{ ft/s}
$$[/tex]
The acceleration due to gravity is given as:
[tex]$$
g = 32 \text{ ft/s}^2.
$$[/tex]
To solve for [tex]$h$[/tex], square both sides of the equation:
[tex]$$
v^2 = 2gh.
$$[/tex]
Plug in the known values:
[tex]$$
12^2 = 2 \cdot 32 \cdot h.
$$[/tex]
Calculate [tex]$12^2$[/tex]:
[tex]$$
144 = 64h.
$$[/tex]
Now, solve for [tex]$h$[/tex]:
[tex]$$
h = \frac{144}{64} = 2.25.
$$[/tex]
Thus, the hammer was dropped from a height of
[tex]$$\boxed{2.25 \text{ feet}}.$$[/tex]
Even though the multiple-choice options given were [tex]$18.0$[/tex] feet and [tex]$1.0$[/tex] foot, based on the calculations, the correct height from which the hammer was dropped is [tex]$2.25$[/tex] feet.
[tex]$$
v = \sqrt{2gh}
$$[/tex]
where
- [tex]$v$[/tex] is the final speed (in feet per second),
- [tex]$g$[/tex] is the acceleration due to gravity (in feet per second squared), and
- [tex]$h$[/tex] is the height above the ground (in feet).
Since the hammer hits the floor at [tex]$12 \text{ ft/s}$[/tex], we have:
[tex]$$
v = 12 \text{ ft/s}
$$[/tex]
The acceleration due to gravity is given as:
[tex]$$
g = 32 \text{ ft/s}^2.
$$[/tex]
To solve for [tex]$h$[/tex], square both sides of the equation:
[tex]$$
v^2 = 2gh.
$$[/tex]
Plug in the known values:
[tex]$$
12^2 = 2 \cdot 32 \cdot h.
$$[/tex]
Calculate [tex]$12^2$[/tex]:
[tex]$$
144 = 64h.
$$[/tex]
Now, solve for [tex]$h$[/tex]:
[tex]$$
h = \frac{144}{64} = 2.25.
$$[/tex]
Thus, the hammer was dropped from a height of
[tex]$$\boxed{2.25 \text{ feet}}.$$[/tex]
Even though the multiple-choice options given were [tex]$18.0$[/tex] feet and [tex]$1.0$[/tex] foot, based on the calculations, the correct height from which the hammer was dropped is [tex]$2.25$[/tex] feet.
