The creation of the moon (a kind of conceptual problem

[Physics122]

Homework Statement

One likely theory for the origin of the Moon is that it was formed by the impact of a Mars-sized object on the young Earth. How fast would the object have to be going when it hit the Earth to vaporize enough rock to make the Moon?

Homework Equations

The Attempt at a Solution

I know I need to make several assumptions for this problem, and that its very conceptual and that as of now the numbers aren't what are important.

I think I'm going to assume that the Earth is only made out of one substance, so that would mean the moon is made out of the same (maybe have it be iron? or silica?)

I think I'm going to need the heat of vaporization of that substance. I think I'll need to know how far away the moon is from the Earth and how large the moon, Mars and Earth is. I believe I'll need to know an estimated temperature of the mars-like object? (room temp??)

Those are all assumptions I need to make but the question is how to piece them together to get the angular velocity of the object coming at earth.

Maybe work backwards to it? Find the distance of the moon to earth, and the size of the moon and know that that big of a chunk must have been rocketed that far from Earth (what equations to get that though?) Moving farther backwards, I'll have to find the heat of vaporization necessary to melt and dis-attach something the size of the moon from Earth and moving even farther backwards from there I can find its angular velocity?

I'm confused as to where to start with this problem and also what equations to use. Numbers aren't important initially, I can compute those later...

 

[Andrew Mason]

Try an angular momentum approach. The angular momentum of the asteroid/earth system before the collision has to equal the angular momentum of the earth/moon system afterward.

The angular momentum of the asteroid before impact is:

$$\vec L_{a} = m_a\vec{v_a}\times \vec{r_e}$$

where r_e is the radius vector from the centre of the Earth to the point of impact.

AM

 

[Physics122]

Try an angular momentum approach. The angular momentum of the asteroid/earth system before the collision has to equal the angular momentum of the earth/moon system afterward.

The angular momentum of the asteroid before impact is:

$$\vec L_{a} = m_a\vec{v_a}\times \vec{r_e}$$

where r_e is the radius vector from the centre of the Earth to the point of impact.

AM

Fantastic. That's a really good way to look at it. I think I'll set r_e pretty far from the center as I believe that the impact occurred near the edge of the earth.

So to find the velocity, I just need to find the angular momentum of the Earth moon system afterwards. But from what I understand, I need to insert the heat of vaporization somewhere in my equation and I don't see how I could relate that to angular momentum?

 

[Physics122]

Try an angular momentum approach. The angular momentum of the asteroid/earth system before the collision has to equal the angular momentum of the earth/moon system afterward.

The angular momentum of the asteroid before impact is:

$$\vec L_{a} = m_a\vec{v_a}\times \vec{r_e}$$

where r_e is the radius vector from the centre of the Earth to the point of impact.

AM

Is this a legit way to go about it?

Initial velocity gives initial kinetic energy KE = 1/2 * Mmars * V^2, where Mmars is the mass of mars.

Some of that energy goes into melting rock. However, depending on the impact, maybe only a small amount of rock was vaporized and the majority of rock was simply knocked away from earth. For a first solution, I could neglect the phase change but maybe I shouldn't?

Now I have a moon-sized set of rocks blasted out of the atmosphere (let's neglect air friction). It's my judgment call about where those rocks came from - were they evenly sprinkled around the Earth's surface so Rinitial=6378.16 kilometers? Were they all from a sphere somewhere inside the Earth's surface so Rinitial<6378.16 kilometers?

Eventually the center of mass of the ejected rocks ended up at the moon's orbital radius. This is a change in potential energy and potential energy, due to the initial kinetic energy (assuming no energy loss due to phase change). KEinitial + PEinitial = PEfinal + KEfinal where PEinitial is the initial potential energy of the ejected rocks, and PEfinal and KEfinal are the energies of the moon in its orbit.

Is that the complete wrong where? maybe I should stick with angular momentum or combine em?

 

[Andrew Mason]

... But from what I understand, I need to insert the heat of vaporization somewhere in my equation and I don't see how I could relate that to angular momentum?

It doesn't. Kinetic energy is not conserved. Much of it goes into ripping matter from the earth. But regardless of the amount of energy lost in the collision, angular momentum has to be conserved. That is why you want to use it, and not energy.

Is this a legit way to go about it?

Yes. Angular momentum is always conserved (assuming that all the matter in the asteroid and Earth before the collision ended up in the moon and the Earth afterward). Energy is not conserved in an inelastic collision such as this.

AM

 

[Phrak]

The prevailing theory is a double impact isn't it?

btw, the mass of Mars is

 

[ideasrule]

The way I understand the question, it's asking how fast the object has to go to vaporize enough rock. It doesn't ask you to account for the energy needed to throw the rock into orbit.