To determine on which planet a space probe of mass 250 kg falling from a height of 25 meters would achieve the highest final speed, we can apply the principles of physics related to free fall. Ignoring air resistance, the final speed [tex]\(v\)[/tex] of the probe can be calculated using the kinematic equation, which derives from the conservation of mechanical energy:
[tex]\[ v = \sqrt{2 \cdot g \cdot d} \][/tex]
where:
- [tex]\( g \)[/tex] is the acceleration due to gravity on the planet.
- [tex]\( d \)[/tex] is the distance fallen (25 meters in this case).
Let's go through the calculation of the final speed for each planet one by one using the given acceleration due to gravity values:
### 1. Venus
[tex]\( g_{Venus} = 8.9 \, m/s^2 \)[/tex]
[tex]\[ v = \sqrt{2 \cdot 8.9 \cdot 25} \][/tex]
The calculated speed is approximately [tex]\( 21.10 \, m/s \)[/tex].
### 2. Earth
[tex]\( g_{Earth} = 9.8 \, m/s^2 \)[/tex]
[tex]\[ v = \sqrt{2 \cdot 9.8 \cdot 25} \][/tex]
The calculated speed is approximately [tex]\( 22.14 \, m/s \)[/tex].
### 3. Uranus
[tex]\( g_{Uranus} = 8.7 \, m/s^2 \)[/tex]
[tex]\[ v = \sqrt{2 \cdot 8.7 \cdot 25} \][/tex]
The calculated speed is approximately [tex]\( 20.86 \, m/s \)[/tex].
### 4. Saturn
[tex]\( g_{Saturn} = 9.0 \, m/s^2 \)[/tex]
[tex]\[ v = \sqrt{2 \cdot 9.0 \cdot 25} \][/tex]
The calculated speed is approximately [tex]\( 21.21 \, m/s \)[/tex].
### Comparison of Speeds
To find the maximum speed, we compare the final speeds calculated for each planet:
- Venus: [tex]\( 21.10 \, m/s \)[/tex]
- Earth: [tex]\( 22.14 \, m/s \)[/tex]
- Uranus: [tex]\( 20.86 \, m/s \)[/tex]
- Saturn: [tex]\( 21.21 \, m/s \)[/tex]
The highest speed attained is [tex]\( 22.14 \, m/s \)[/tex] on Earth.
### Conclusion
Thus, the space probe would have the highest speed after falling 25 meters on Earth, reaching approximately [tex]\( 22.14 \, m/s \)[/tex].
