To determine on which planet the space probe would achieve the highest speed after falling 25 meters, we can use the formula for the final velocity of an object in free fall:
[tex]\[ v = \sqrt{2 \cdot g \cdot h} \][/tex]
where:
- [tex]\( v \)[/tex] is the final velocity,
- [tex]\( g \)[/tex] is the acceleration due to gravity on the planet,
- [tex]\( h \)[/tex] is the distance fallen.
Let's analyze each planet step by step:
1. Venus:
[tex]\[ g = 8.9 \, m/s^2 \][/tex]
[tex]\[ h = 25 \, m \][/tex]
[tex]\[ v_{Venus} = \sqrt{2 \cdot 8.9 \cdot 25} \][/tex]
[tex]\[ v_{Venus} = \sqrt{445} \][/tex]
[tex]\[ v_{Venus} \approx 21.095 \, m/s \][/tex]
2. Earth:
[tex]\[ g = 9.8 \, m/s^2 \][/tex]
[tex]\[ h = 25 \, m \][/tex]
[tex]\[ v_{Earth} = \sqrt{2 \cdot 9.8 \cdot 25} \][/tex]
[tex]\[ v_{Earth} = \sqrt{490} \][/tex]
[tex]\[ v_{Earth} \approx 22.136 \, m/s \][/tex]
3. Uranus:
[tex]\[ g = 8.7 \, m/s^2 \][/tex]
[tex]\[ h = 25 \, m \][/tex]
[tex]\[ v_{Uranus} = \sqrt{2 \cdot 8.7 \cdot 25} \][/tex]
[tex]\[ v_{Uranus} = \sqrt{435} \][/tex]
[tex]\[ v_{Uranus} \approx 20.857 \, m/s \][/tex]
4. Saturn:
[tex]\[ g = 9.0 \, m/s^2 \][/tex]
[tex]\[ h = 25 \, m \][/tex]
[tex]\[ v_{Saturn} = \sqrt{2 \cdot 9.0 \cdot 25} \][/tex]
[tex]\[ v_{Saturn} = \sqrt{450} \][/tex]
[tex]\[ v_{Saturn} \approx 21.213 \, m/s \][/tex]
By comparing the final speeds:
- Venus: 21.095 m/s
- Earth: 22.136 m/s
- Uranus: 20.857 m/s
- Saturn: 21.213 m/s
We can see that the highest final velocity after falling 25 meters is on Earth, with a speed of approximately 22.136 m/s.
Therefore, the space probe would have the highest speed after falling 25 meters on Earth.
The correct answer is:
A. Earth
