To find the vertical height from which Jeff dropped the bottle, let's follow these steps:
1. Given Data:
- Mass of the bottle, [tex]\( m = 0.25 \)[/tex] kilograms
- Velocity when it hits the ground, [tex]\( v = 14 \)[/tex] meters/second
- Acceleration due to gravity, [tex]\( g = 9.8 \)[/tex] meters/second[tex]\(^2\)[/tex]
2. Calculate Kinetic Energy (KE):
- The formula for kinetic energy is [tex]\( KE = \frac{1}{2} m v^2 \)[/tex]
- Substituting the given values:
[tex]\[
KE = \frac{1}{2} \times 0.25 \, \text{kg} \times (14 \, \text{m/s})^2
\][/tex]
[tex]\[
KE = \frac{1}{2} \times 0.25 \times 196
\][/tex]
[tex]\[
KE = \frac{1}{2} \times 49
\][/tex]
[tex]\[
KE = 24.5 \, \text{Joules}
\][/tex]
3. Relation Between Kinetic Energy and Potential Energy:
- When the bottle is at the height [tex]\( h \)[/tex] and is about to be dropped, it has potential energy (PE) equal to the kinetic energy (KE) it has just before hitting the ground (assuming no energy losses).
- The formula for potential energy is [tex]\( PE = m \times g \times h \)[/tex]
- Since [tex]\( PE = KE \)[/tex], we have:
[tex]\[
m \times g \times h = KE
\][/tex]
[tex]\[
0.25 \times 9.8 \times h = 24.5
\][/tex]
4. Solve for Height (h):
[tex]\[
2.45 h = 24.5
\][/tex]
[tex]\[
h = \frac{24.5}{2.45}
\][/tex]
[tex]\[
h = 10 \, \text{meters}
\][/tex]
So, the vertical height from which Jeff dropped the bottle is [tex]\( 10 \)[/tex] meters.
Hence, the correct answer to fill in the box is:
[tex]\[
\boxed{10}
\][/tex]
