Let's solve this problem step-by-step.
### Given Data:
- Mass of the box ([tex]\( m \)[/tex]) = 2 kg
- Initial kinetic energy ([tex]\( KE_{initial} \)[/tex]) = 2 Joules
- Initial height ([tex]\( h_{initial} \)[/tex]) = 10 meters
- Final height ([tex]\( h_{final} \)[/tex]) = 5 meters
- Acceleration due to gravity ([tex]\( g \)[/tex]) = 9.81 m/s[tex]\(^2\)[/tex]
### Calculation:
Step 1: Calculate Initial Potential Energy
[tex]\( PE_{initial} = m \cdot g \cdot h_{initial} \)[/tex]
[tex]\[
PE_{initial} = 2 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 10 \, \text{m}
\][/tex]
[tex]\[
PE_{initial} = 196.2 \, \text{Joules}
\][/tex]
Step 2: Calculate Final Potential Energy
[tex]\( PE_{final} = m \cdot g \cdot h_{final} \)[/tex]
[tex]\[
PE_{final} = 2 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 5 \, \text{m}
\][/tex]
[tex]\[
PE_{final} = 98.1 \, \text{Joules}
\][/tex]
Step 3: Calculate Change in Potential Energy
[tex]\[
\Delta PE = PE_{initial} - PE_{final}
\][/tex]
[tex]\[
\Delta PE = 196.2 \, \text{Joules} - 98.1 \, \text{Joules}
\][/tex]
[tex]\[
\Delta PE = 98.1 \, \text{Joules}
\][/tex]
Step 4: Calculate Final Kinetic Energy
The change in potential energy results in a change in kinetic energy:
[tex]\[
KE_{final} = KE_{initial} + \Delta PE
\][/tex]
[tex]\[
KE_{final} = 2 \, \text{Joules} + 98.1 \, \text{Joules}
\][/tex]
[tex]\[
KE_{final} = 100.1 \, \text{Joules}
\][/tex]
This concludes part (a):
(a) The kinetic energy when the box is 5 meters above the ground is 100.1 J.
Step 5: Calculate the Speed of the Box at 5 meters above the Ground:
Using the kinetic energy formula:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
Solving for [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{\frac{2 \cdot KE}{m}}
\][/tex]
Substitute [tex]\( KE_{final} \)[/tex] and [tex]\( m \)[/tex]:
[tex]\[
v = \sqrt{\frac{2 \cdot 100.1 \, \text{Joules}}{2 \, \text{kg}}}
\][/tex]
[tex]\[
v = \sqrt{\frac{200.2}{2}}
\][/tex]
[tex]\[
v = \sqrt{100.1}
\][/tex]
[tex]\[
v \approx 10.005 \, \text{m/s}
\][/tex]
This concludes part (b):
(b) The speed of the box when it is 5 meters above the ground is approximately 10.005 m/s.
