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 8.93×1022 kg and a radius of 1821 km . For this calculation, ignore any variation in gravity over the 500-km range of the debris.How high would this material go on earth if it were ejected with the same speed as on Io?(Express your answer to three significant figures and include the appropriate units.)

Answer 1

Answer:

91.5 km

Explanation:

Hi!

If we are to ignore the variation in gravity we can use the formula for teh potential energy near the surface of a planet:

mgh

If the energy of the material ejected from the volcano on Io's surface is the same on earth's surface we have:

$mg_{Io}h_{Io} =mg_{e}h_{e}$

where subindexes Io and e means Jupiter's moon Io, and Earth, respectively

solving for h_e

$h_{e} = h_{Io} \frac{g_{e}}{g_{Io}}$

The acceleration due to the gravity of a planet can be calculated as:

$g = G \frac{m}{R^2}$

Where R and m are the radius  and mass of the planet

Therefore:

$h_{e} = h_{Io} (\frac{m_{Io}}{m_e})(\frac{R_e}{R_{Io}})^2$

m_e = 5,972 × 10^24 kg

R_e = 6 371 km

Replacing all given values:

h_e = 500 km *(0.183) = 91.515 990 km