To determine which planet would cause the space probe to have the highest speed after falling 25 meters, we need to calculate the final speed for the space probe on each planet using the following formula:
[tex]\[ v = \sqrt{2 \times g \times h} \][/tex]
where:
- [tex]\( v \)[/tex] is the final speed.
- [tex]\( g \)[/tex] is the acceleration due to gravity on the planet (in [tex]\(\text{m/s}^2\)[/tex]).
- [tex]\( h \)[/tex] is the distance fallen (in meters).
Given the acceleration due to gravity for each planet and the distance of 25 meters, we can calculate the final speed for each planet.
Here is the data we have:
- Venus: [tex]\( g = 8.9 \text{ m/s}^2 \)[/tex]
- Earth: [tex]\( g = 9.8 \text{ m/s}^2 \)[/tex]
- Uranus: [tex]\( g = 8.7 \text{ m/s}^2 \)[/tex]
- Saturn: [tex]\( g = 9.0 \text{ m/s}^2 \)[/tex]
Now we perform the calculations:
1. For Venus:
[tex]\[ v = \sqrt{2 \times 8.9 \times 25} \approx 21.10 \text{ m/s} \][/tex]
2. For Earth:
[tex]\[ v = \sqrt{2 \times 9.8 \times 25} \approx 22.14 \text{ m/s} \][/tex]
3. For Uranus:
[tex]\[ v = \sqrt{2 \times 8.7 \times 25} \approx 20.86 \text{ m/s} \][/tex]
4. For Saturn:
[tex]\[ v = \sqrt{2 \times 9.0 \times 25} \approx 21.21 \text{ m/s} \][/tex]
To determine the planet with the highest final speed:
- Venus: [tex]\( \approx 21.10 \text{ m/s}\)[/tex]
- Earth: [tex]\( \approx 22.14 \text{ m/s}\)[/tex]
- Uranus: [tex]\( \approx 20.86 \text{ m/s}\)[/tex]
- Saturn: [tex]\( \approx 21.21 \text{ m/s}\)[/tex]
Among these values, the highest speed is [tex]\( 22.14 \text{ m/s} \)[/tex] on Earth.
Therefore, the planet on which the space probe would have the highest speed after falling 25 meters is:
B. Earth
