19. A space probe is flying by Jupiter (M = 1.9 x 1027 kg) at a distance of 400,000 km from the surface. Jupiter's radius is 7.15 x 107 m. How fast would the space probe have to be traveling in order to escape the gravity of Jupiter?
19. A space probe is flying by Jupiter (M = 1.9 x 1027 kg) at a distance of 400,000 km from the surface. Jupiter's radius is 7.15 x 107 m. How fast would the space probe have to be traveling in order to escape the gravity of Jupiter?
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Chapter1: Units, Trigonometry. And Vectors
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![**Problem 19: Calculating Escape Velocity of a Space Probe Near Jupiter**
A space probe is flying by Jupiter, which has a mass (M) of \(1.9 \times 10^{27}\) kg. The probe is at a distance of 400,000 km from the surface of Jupiter. Jupiter's radius is \(7.15 \times 10^7\) meters. The question is to determine how fast the space probe would have to be traveling in order to escape the gravity of Jupiter.
**Explanation:**
To solve this problem, you need to use the formula for escape velocity:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
Where:
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11}\, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2}\))
- \( M \) is the mass of the celestial body, in this case, Jupiter
- \( R \) is the distance from the center of the celestial body to the object
Given:
- \( M = 1.9 \times 10^{27} \, \text{kg} \)
- Radius of Jupiter = \(7.15 \times 10^7 \, \text{m} \)
- Distance from surface = 400,000 km = \(4.0 \times 10^8 \, \text{m} \)
Total distance \( R \) from the center of Jupiter:
\[ R = \text{radius of Jupiter} + \text{distance from surface} = 7.15 \times 10^7 \, \text{m} + 4.0 \times 10^8 \, \text{m} \]
Use these values in the escape velocity formula to find the required speed for the space probe to escape Jupiter's gravity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8288b48-397c-47a1-92e8-4d9421f21f30%2F309df288-81dd-4618-b3e5-c4a2be9cf0e8%2Fsq128un_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 19: Calculating Escape Velocity of a Space Probe Near Jupiter**
A space probe is flying by Jupiter, which has a mass (M) of \(1.9 \times 10^{27}\) kg. The probe is at a distance of 400,000 km from the surface of Jupiter. Jupiter's radius is \(7.15 \times 10^7\) meters. The question is to determine how fast the space probe would have to be traveling in order to escape the gravity of Jupiter.
**Explanation:**
To solve this problem, you need to use the formula for escape velocity:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
Where:
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11}\, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2}\))
- \( M \) is the mass of the celestial body, in this case, Jupiter
- \( R \) is the distance from the center of the celestial body to the object
Given:
- \( M = 1.9 \times 10^{27} \, \text{kg} \)
- Radius of Jupiter = \(7.15 \times 10^7 \, \text{m} \)
- Distance from surface = 400,000 km = \(4.0 \times 10^8 \, \text{m} \)
Total distance \( R \) from the center of Jupiter:
\[ R = \text{radius of Jupiter} + \text{distance from surface} = 7.15 \times 10^7 \, \text{m} + 4.0 \times 10^8 \, \text{m} \]
Use these values in the escape velocity formula to find the required speed for the space probe to escape Jupiter's gravity.
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