Help with Contact Mechanics between Two Spheres

[aricleather]

Hi. I am currently in Grade 7, and am doing an inquiry project in math.

 

Our current math subject is probability. We were told to pick a question, and using multiple subquestions, calculate the theoretical probability of the event.

 

Me and my partner are big on video games, so we took a classic theme, Majora's Mask. In the game, the moon is going to crash onto the planet and destroy everything. Majora's moon is rather small, so we decided we would use the real moon in our question:

 

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

 

We have lots of the little questions down, like:

 

What is the probability of the moon knocking Earth out of it's orbit.

What is the probability of the moon blocking out the sun?

etc.

 

But now that the project is due in 3 days, we are stuck on the toughest question:

 

What is the probability of being in the area the moon lands?

 

So we did a little google search on contact mechanics between spheres, and we got this formula from Wikipedia:

 

I'm sure we could've solved it on our own, but we had no way of knowing that this was the proper formula, and nowhere in the article were the variables defined.

 

So, can anyone assist us in finding the contact area between the moon and the Earth?

 

Edit: Considering both the moon and the Earth are perfect spheres.

Edited May 21, 2014 by aricleather

[studiot]

Contact Mechanics?

7th grade?

 

Gosh that's wa...aay too complicated.

 

I think a knowledge of Pythagoras will suffice.

 

 

Not to mention that it is wholely inappropriate for your question.

Contact mechanics refers to the contact of solid bodies, and the Earth has a lot of water to land on.

 

 

When you do mathematical modelling, which is what you are attempting, you make assumptions as well as gathering data.

 

Required data.

 

Radius of Moon

Radius of Earth

Total population of earth.

Population density map, you can measure areas on. This last part is important since maps always distort something and you need one that does not distort area.

% of Earth's surface covered by water.

 

Now I am going to assume that the impact will penetrate a depth of 1.5 kilometres or 1500 metres. If the impact is on land, not water.

We can test the validy of this assumption later and alter it if necessary, but is is reasonable.

 

The emboldened condition gives us our first statistical question/calculation.

The % of the earth's suface that is land gives a (partial) probability that the imact will hit land and penetrate my 1500m

 

Now look at my diagram. I have left some data for you to find out.

 

 

You can see the impact zone where the Moon hits as the shaded area above the dashed line.

 

The diagram also shows some simple geometry that will calculate the radius of this flattened area, using Pythagoras.

 

I make that radius 138320 metres. You will need to calculate the area.

 

This is a good deal less than the radius of the Moon which gives the radius of largest flattened area that could be made.

 

So you have an estimate of the flattened area.

 

Now your chances of being flattened depends upon the population density where the Moon hits.

Obviously a strike on top of the Himalayas or central Australia will not kill as many people as a direct hit on the east coast of the USA.

So divide you population densities into bands and get an area for each band from your population density map.

These areas can be converted into (partial) probabilities of a kill.

 

How does this sound for a working model?

Run it past your tutor.

Edited May 21, 2014 by studiot

[imatfaal]

Hi. I am currently in Grade 7, and am doing an inquiry project in math.

 

Our current math subject is probability. We were told to pick a question, and using multiple subquestions, calculate the theoretical probability of the event.

 

Me and my partner are big on video games, so we took a classic theme, Majora's Mask. In the game, the moon is going to crash onto the planet and destroy everything. Majora's moon is rather small, so we decided we would use the real moon in our question:

 

 

Hey start with the easy ones? :-D

 

What is the probability of being in the area the moon lands?

 

Depends on how far it embeds itself and the size of the crater.

 

In fact that wiki page does give you all the variables it uses if you read it carefully - But you will need to do some research to get figures. Not sure what the point is though - you are talking end of the world as we know it