Question

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93×1022kg and a radius of 1821 km.

Part A
How high would this material go on earth if it were ejected with the same speed as on Io? (RE = 6370 km, mE=5.96×1024kg)
Express your answer with the appropriate units.
View Available Hint(s)for Part A
Hint 1for Part A. Find speed of the material ejected from Io.
Step 1: Use conservation of energy (universal potential energy) to find the speed with which the material is ejected from Io. Remember that at the maximum altiute reached bu the material the speed is zero.

Step 2: After you find the speed of ejection from step 1, use the conservation of energy for the material ejected with this speed from Earth and find the altitude (distance from surface).

Answer 1

To determine the altitude on Earth, we first calculate the eruption ejection speed on Io using conservation of energy, assuming that potential energy at Io's surface plus kinetic energy is equal to potential energy at maximum altitude. This speed is then applied to Earth's gravitational potential energy equations to find how high material would reach on Earth.

Step-by-step explanation:

Calculating Ejection Speed and Altitude

To determine the maximum altitude that volcanic material would reach if ejected from Earth with the same initial speed as on Io, we first need to calculate the ejection speed on Io using the concept of conservation of energy. By assuming that the total mechanical energy (kinetic plus potential) at the surface level is equal to the potential energy at the maximum altitude, we find the initial speed. We then apply this speed to Earth's stronger gravitational field to determine how high the material would go here.

Step 1: Use conservation of energy to find the ejection speed on Io. The gravitational potential energy at the surface plus the kinetic energy must equal the potential energy at the maximum altitude where kinetic energy is zero.

Potential Energy on Io's surface: PEi = - G Mm / Ri
Kinetic Energy on Io's surface: KEi = (1/2) mv²
Gravitational Potential Energy at max altitude on Io: PEi_max = - G Mm / (Ri + h)

Conservation of Energy: KEi + PEi = PEi_max
Solving for v gives us the ejection speed on Io.

Step 2: Using the ejection speed found from Io, apply it to Earth conditions to find the altitude on Earth.

Potential Energy on Earth's surface: PEe = - G Me m / Re
Kinetic Energy on Earth's surface: KEe = (1/2) mv² (using the same v from Io)
Gravitational Potential Energy at max altitude on Earth: PEe_max = - G Me m / (Re + h')

Conservation of Energy: KEe + PEe = PEe_max
Solving for h' gives us the maximum altitude that material would reach if ejected from Earth.