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DameLight

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## Homework Statement

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 300 km (or even higher) above the surface. Io has a mass of 8.94 × 10

^{22}kg and a radius of 1815 km . Ignore any variation in gravity over the 300 km range of the debris.

h = 300 km = 300 * 10

^{3}m

m

_{Io}= 8.94 × 10

^{22}kg

r

_{Io}= 1815 km = 1815 * 10

^{3}m

How high would this material go on Earth if it were ejected with the same speed as on Io?

M

_{E}= 5.97 × 10

^{24}

r

_{E}= 6371 km = 6371 * 10

^{3}m

G = 6.67 × 10

^{−11}

## Homework Equations

U

_{g}= - (GMm)/r

F

_{g}= (GMm)/r

^{2}

## The Attempt at a Solution

*Conservation of ME on Io*

PE = PE + KE

- (GMm)/(r+h) = - (GMm)/r + 1/2 mv

^{2}

- (GMm)/(r+h) + - (GMm)/r = 1/2 v

^{2}

2(- (GM)/(r+h) + - (GM)/r)) = v

^{2}

v = √(2(- (GM)/(r+h) + - (GM)/r)))

v = √(2(- (6.67 × 10

^{−11}* 8.94 × 10

^{22})/(1815 * 10

^{3}+ 300 * 10

^{3}) + - ((6.67 × 10

^{−11}* 8.94 × 10

^{22})/1815 * 10

^{3})))

v = 2563.36 m/s

*Conservation of ME on Earth*

PE = PE + KE

- (GMm)/(r+h) = - (GMm)/r + 1/2 mv

^{2}

(- (GMm)/r + 1/2 mv

^{2})/- (GMm) = (r+h)

h = (- (GMm)/r + 1/2 mv

^{2})/- (GMm) - r

h = (- (6.67 × 10

^{−11}* 5.97 × 10

^{24})/6371 * 10

^{3}+ 1/2 * 2563.36

^{2})/- (6.67 × 10

^{−11}* 5.97 × 10

^{24}) - 6371 * 10

^{3}

h = 6.37 * 10

^{6}

Points Changed:

1. Initial PE should exist

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