Which celestial body would have the strongest gravitational pull on a satellite orbiting 100 km above its surface?
Explain your answer.
Answer (1)
Calculate the radii of Earth, Jupiter, and Mars: R E = 6371000 m, R J = 69911000 m, R M = 3389500 m.
Calculate the distances from the center of each celestial body to the satellite: r E = 6471000 m, r J = 70011000 m, r M = 3489500 m.
Calculate r 2 M for each celestial body: Earth: 1.4257 × 1 0 11 , Jupiter: 3.8763 × 1 0 11 , Mars: 5.2724 × 1 0 10 .
Compare the values and determine that Jupiter has the strongest gravitational pull: Jupiter .
Explanation
Understanding the Problem We are given the masses of Earth, Jupiter, and Mars, and we need to determine which celestial body exerts the strongest gravitational pull on a satellite orbiting 100 km above its surface. The gravitational force is given by F = G r 2 M m , where G is the gravitational constant, M is the mass of the celestial body, m is the mass of the satellite, and r is the distance between the center of the celestial body and the satellite. Since G and m are constant, we only need to compare the values of r 2 M for each celestial body.
Calculating Distances First, we need to find the radii of Earth, Jupiter, and Mars. The radii are approximately:
Earth: R E = 6371 km = 6371000 m
Jupiter: R J = 69911 km = 69911000 m
Mars: R M = 3389.5 km = 3389500 m
The satellite orbits 100 km above the surface, which is equal to 100000 m. Therefore, the distances from the center of each celestial body to the satellite are:
Earth: r E = 6371000 + 100000 = 6471000 m
Jupiter: r J = 69911000 + 100000 = 70011000 m
Mars: r M = 3389500 + 100000 = 3489500 m
Calculating M/r^2 Now, we calculate the values of r 2 M for each celestial body:
Earth: r E 2 M E = ( 6471000 ) 2 5.97 × 1 0 24 ≈ 1.4257 × 1 0 11 kg/m 2
Jupiter: r J 2 M J = ( 70011000 ) 2 1.90 × 1 0 27 ≈ 3.8763 × 1 0 11 kg/m 2
Mars: r M 2 M M = ( 3489500 ) 2 6.42 × 1 0 23 ≈ 5.2724 × 1 0 10 kg/m 2
Comparing Gravitational Pulls Comparing the values of r 2 M , we have:
Earth: 1.4257 × 1 0 11
Jupiter: 3.8763 × 1 0 11
Mars: 5.2724 × 1 0 10
Since 3.8763 × 1 0 11 is the largest value, Jupiter has the strongest gravitational pull on the satellite.
Final Answer Therefore, Jupiter would have the strongest gravitational pull on a satellite orbiting 100 km above its surface.
Examples
Understanding gravitational pull is crucial in space exploration. For example, when planning a mission to Mars, scientists need to calculate the precise trajectory of the spacecraft. This involves accounting for the gravitational forces of the Earth, the Sun, and Mars itself. By accurately modeling these forces, engineers can ensure the spacecraft arrives at its destination efficiently and safely. Similarly, understanding the gravitational forces of Jupiter is essential for missions exploring its moons, like Europa, where scientists suspect there may be subsurface oceans.
