Question

Solo, a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over 500 km high. Due to the satellite's small mass, the acceleration due to gravity on Io is only 1.81 m/s^2, and solo has no appreciable atmosphere. Assume that there is no variation in gravity over the distance traveled.

a) What must be the speed of material just as it leaves the volcano to reach an altitude of 440 km (answer in m/s)

b) If the gravitational potential energy is zero at the surface, what is the potential energy for a 26 kg fragment at its maximum height (500 km) on solo? (answer in joules)

c) How much would this gravitational potential energy be if it were at the same height (500 km) above earth? (answer in joules)

Answer 1

A) To find the speed of the material just as it leaves the volcano, we can use the conservation of energy principle. The initial potential energy of the material is zero (since it's at the surface of the moon), and the final potential energy is given by mgh, where m is the mass of the material, g is the acceleration due to gravity, and h is the height above the surface. The final kinetic energy of the material is given by 1/2mv^2, where v is the velocity of the material.

Since initial potential energy + initial kinetic energy = final potential energy + final kinetic energy, we can set up the following equation:

0 = mgh + 1/2mv^2

where

m = 26 kg (mass of the fragment)

g = 1.81 m/s^2 (acceleration due to gravity on Io)

h = 440 x 10^3 m (height of the volcano)

Solving for v, we get:

v = sqrt(2mgh) = sqrt(2 * 26 kg * 1.81 m/s^2 * 440 x 10^3 m) = 791.4 m/s

B) To find the potential energy of the fragment at its maximum height, we use the formula for gravitational potential energy:

PE = mgh

where

m = 26 kg (mass of the fragment)

g = 1.81 m/s^2 (acceleration due to gravity on Io)

h = 500 x 10^3 m (maximum height of the volcano)

PE = 26 kg * 1.81 m/s^2 * 500 x 10^3 m = 2.295 x 10^7 J

C) To find the potential energy of the fragment if it were at the same height above the Earth, we use the same formula and the gravitational constant of the Earth:

PE = mgh

where

m = 26 kg (mass of the fragment)

g = 9.81 m/s^2 (acceleration due to gravity on Earth)

h = 500 x 10^3 m (maximum height of the volcano)

PE = 26 kg * 9.81 m/s^2 * 500 x 10^3 m = 1.305 x 10^8 J

It's important to notice that the gravitational constant of the Earth is about 4.4 times greater than the gravitational constant of Io. That's why the gravitational potential energy of the fragment is much greater if it were at the same height above the Earth.