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. 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?

Answer 1

Answer:

91.54 km

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

r = Radius of Io = 1821000 m

m = Mass of the Io =  8.93 × 10²² kg

$H_i$ = 500000 m

g = Acceleration due to gravity on Earth = 9.81 m/s²

Acceleration due to gravity on Io

$g_i=\frac{GM}{r^2}\\\Rightarrow g_i=\frac{6.67\times 10^{-11}\times 8.93\times 10^{22}}{1821000^2}\\\Rightarrow g_i=1.796\ m/s^2$

$H_e=\frac{v^2}{2g}$

$H_i=\frac{v^2}{2g_l}$

Dividing the above equations we get

$\frac{H_e}{H_i}=\frac{\frac{v^2}{2g}}{\frac{v^2}{2g_i}}\\\Rightarrow \frac{H_e}{H_i}=\frac{g_i}{g}\\\Rightarrow \frac{H_e}{H_i}=\frac{1.796}{9.81}\\\Rightarrow H_e=H_i\times \frac{1.796}{9.81}\\\Rightarrow H_e=500000\times \frac{1.796}{9.81}\\\Rightarrow H_e=91539.245\ m=91.54\ km$

The material would go 91.54 km on earth if it were ejected with the same speed as on Io.