Several volcanoes have been observed erupting on the surface of Jupiter's closest moon, Io.?

Question

Several volcanoes have been observed erupting on the surface of Jupiter's closest moon, Io. Suppose that material ejected from one of these volcanoes reaches a height of 4.20 km after being projected straight upward with an initial speed of 240 m/s.
(a) Given that the radius of Io is 3303 km, outline a strategy that allows you to calculate the mass of Io.

(b) Use your strategy to calculate Io's mass.
kg

Answer 1

The acceleration of gravity near the surface of Io can be computed from Newton's Law of Gravity:

g = G M / r^2

where G is the Newton gravity constant, M the mass of Io, and r the radius of Io.

You can also find g from kinematics using the information given.

2 g h = v^2 - vo^2

g = acceleration of gravity near the surface of Io (unknown)
h = height reached by ejected material (given)
v = final velocity of the ejected material (zero)
vo = initial velocity of ejected material (given)

So solve for g, then use Newton's Law of Gravity to find the mass of Io.

Answer 2

There are going to be two steps to this. First, we need to find out the acceleration due to gravity. Next, we will use that knowledge to solve for the mass of Io.

The first part is all kinematics. Sum the forces, and solve for a.

m*a=m*-g
(note, g does not equal 9.81 here. g is due to Io's gravity)
Drop out the masses, and you have
a=-g
Now, integrate a few times,
v(t)= vo-g*t
x(t)=vo*t-g*t^2/2

vo is equal to 240 m/s, and now we solve for a.
v(t)=0 when height is equal to 4200 meters.
0=vo-g*t
t=vo/g
Plug it in for x(t)
4200=vo^2/g - vo^2/(2*g)
g=vo^2/8400

Now that we have a known for the acceleration due to gravity, we can solve for the mass of the planet.
Recall that the force due to gravity is such that
Fg(1on2)=m1*a=(G*m1*m2)/r^2

where G is the gravitational constant 6.673*10^-12 (I think. Double check this one),
m1 is the mass of the rock
m2 is the mass of the planet
and r is the center to center distance (radius of the planet)

We can cancel out the m1's right away, and plug in g for a

vo^2/8400= G*m2/r^2

and you wind up with
m2= (vo^2*r^2)/(8400*G)

This should be it, unless I did some bad math. Double check for yourself, but these are the steps needed.