Jupiter\'s moon Io has active volcanoes (in fact, it is the most volcanically ac

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 550 km  (or even higher) above the surface. Io has a mass of 8.94×1022 kg  and a radius of 1815 km. Ignore any possible air resistance.

A) Use conservation of energy to solve for the initial speed of the ejected material (assume it leaves from the surface of Io) such that it reaches a maximum height of 550 km above the surface.

B) Now assume that the same volcano existed on Earth. Use conservation of energy to solve for the maximum height above the Earth’s surface that the ejected material would travel. Use 5.98×1024 kg for the mass of the Earth and 6.38×106 m  for the radius of the Earth. Again ignore any air resistance. Express your answer in kilometers (km)

Explanation / Answer

let m is the mass of volacano

A) Apply conservation of energy

(1/2)*m*v^2 - G*M*m/R = G*M*m/(R+h)

(1/2)*v^2 - G*M/R = -G*M/(R+h)

(1/2)*v^2 = G*M/R - G*M/(R+h)

v = sqrt(2)*sqrt(G*M/R - G*M/(R+h))

= sqrt(2)*sqrt(6.67*10^-11*8.94*10^22/(1815*10^3) - 6.67*10^-11*8.94*10^22/(1815*10^3 + 550*10^3))

= 1236 m/s or 1.236 km/s

B) A) Apply conservation of energy

(1/2)*m*v^2 - G*Me*m/Re = G*Me*m/(Re+h)

(1/2)*v^2 - G*Me/R = -G*Me/(Re+h)

(1/2)*v^2 = G*Me/R - G*Me/(Re+h)

v = sqrt(2)*sqrt(G*Me/Re - G*Me/(Re+h))

= sqrt(2)*sqrt(6.67*10^-11*5.98*10^24/(6380*10^3) - 6.67*10^-11*5.98*10^24/(6380*10^3 + 550*10^3))

= 3150 m/s or 3.15 km/s

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