Help with Contact Mechanics Between two Spheres

[aricleather]

Homework Statement

Hello. Me and my partner are in Grade 7, and are currently doing an inquiry project in Math. We were told to take a question, and calculate theoretical probability of both outcomes. In this case, it would be surviving or dying. Our question is:

What is the probability of surviving if the moon struck Earth at 3,683km/h?

We had a lot of smaller questions that were easily answered, and 4 days from the due date, we get stuck on this.

Since we're using the moon, and the Earth, we have this data:

Surface area of Earth: 510,072,000 km²
Surface area of the Moon: 37,930,000 km²

Homework Equations

We googled "contact area between to spheres", and Wikipedia had a formula:

But we have no way of telling if the formula is correct. Also, the variables aren't defined anywhere in the entry.

The Attempt at a Solution

Honestly, we couldn't even get here. The first thing we tried was diving the Earth's surface area by the moon's, but than we realized that not all of the moon will be touching the Earth, and that's where contact mechanics come in.

Can anyone help?

 

[Dick]

Homework Statement

Hello. Me and my partner are in Grade 7, and are currently doing an inquiry project in Math. We were told to take a question, and calculate theoretical probability of both outcomes. In this case, it would be surviving or dying. Our question is:

What is the probability of surviving if the moon struck Earth at 3,683km/h?

We had a lot of smaller questions that were easily answered, and 4 days from the due date, we get stuck on this.

Since we're using the moon, and the Earth, we have this data:

Surface area of Earth: 510,072,000 km²
Surface area of the Moon: 37,930,000 km²

Homework Equations

We googled "contact area between to spheres", and Wikipedia had a formula:

But we have no way of telling if the formula is correct. Also, the variables aren't defined anywhere in the entry.

The Attempt at a Solution

Honestly, we couldn't even get here. The first thing we tried was diving the Earth's surface area by the moon's, but than we realized that not all of the moon will be touching the Earth, and that's where contact mechanics come in.

Can anyone help?

That's probably not really relevant physics to what would happen if the moon were to impact the earth. Treating them as solid spheres that would roll along each other isn't what would happen. You might think about looking up Roche limit, or estimating the total kinetic energy that would be released if this happened. Compare it with the energy released by the impact that's credited with killing the dinosaurs. Actually, a similar impact is believed to have occurred early in Earth's history. It created the moon. Try and look that up.

 

[aricleather]

That's probably not really relevant physics to what would happen if the moon were to impact the earth. Treating them as solid spheres that would roll along each other isn't what would happen. You might think about looking up Roche limit, or estimating the total kinetic energy that would be released if this happened. Compare it with the energy released by the impact that's credited with killing the dinosaurs. Actually, a similar impact is believed to have occurred early in Earth's history. It created the moon. Try and look that up.

Forgot to mention that we have an okay to treat them as solid spheres. Don't want to get into that stuff during Grade 7 Anyway, we've looked into the giant impact hypothesis already and didn't find anything that useful. There has to be some sort of formula to calculate the contact area between two spheres?

 

[Dick]

Forgot to mention that we have an okay to treat them as solid spheres. Don't want to get into that stuff during Grade 7 Anyway, we've looked into the giant impact hypothesis already and didn't find anything that useful. There has to be some sort of formula to calculate the contact area between two spheres?

Well, if you've looked into the giant impact stuff already you probably know, realistically, the probability of surviving something like that is 0%. The moon wouldn't survive and the Earth's crust wouldn't survive either.

The formulae you are looking at are for two deformable elastic spheres being pushed together by a known force. And it won't let you compute 'a'. That's an input value. In principal, you could compute it if you knew the elastic properties of the spheres, but you don't and I wouldn't know how to do it if you did. If the spheres were perfectly rigid then 'a' would be zero and the pressure would be infinite. I don't think this is good approach.

 

[haruspex]

Can't add much to Dick's response, but..
You need to decide whether you want (a) to stick with your original challenge (survivability), or (b) to switch to asking what the 'contact' area would be from such a collision.
For (a), it's nothing to do with contact areas. Much more to do with surface conditions around the globe that result - in the first instance the massive pulse of heat, earthquakes, volcanoes, tsunamis; but later (maybe worse) a 'nuclear winter' from all the dust, smoke and soot in the atmosphere.
For (b) it would be totally inaccurate to treat such a collision as elastic. The larger the scale, the more objects behave like soft putty. It would be more accurate to treat them as colliding drops of water.

 

[AlephZero]

I don't know what "grade 7" means in your country (you didn't say which country that is) but cutting the problem down to something that you can calculate, find the kinetic energy of the moon when it hits the earth. Compare that with the energy released by the biggest man-made nuclear weapons that have been tested, or the energy in a storm like Hurricane Katrina. (you can find estimates of those quantities on the web).