Ratio of projectile to ejecta mass, shoot the moon.

[Spinnor]

Say we shoot the surface of the moon with various sized steel balls of size a cm and larger, say up to a km in size. Say the balls hit the surface of the moon moving at 20,000 mph. Is there a simple formula for the mass of the ejecta as a function of the mass of the projectile? Will the formula depend on whether the projectile hits dust or rock or a combination of the two?

Thanks for any help!

 

[Spinnor]

After a bit of Googling, wow, what a find,

http://keith.aa.washington.edu/craterdata/scaling/index.htm

From, Crater Database and Scaling Tools,

http://keith.aa.washington.edu/craterdata/

The math behind the calculator,

http://keith.aa.washington.edu/craterdata/scaling/theory.pdf

 

[Drakkith]

Wow, nice find!

 

[Spinnor]

Used the above calculator, seemed buggy for multiple calculations, and got the following result. For a steel projectile striking the moon at 10,000 mph perpendicularly, the mass of the projectile, M, was roughly proportional to the volume, V, of ejecta for sizes from 1 cm to 10 m,

M ≈ V

above that size

M^α ≈ V

Where α is less then 1.

Is the above an example of scale invariance? If I draw a picture of the projectile next to the volume of ejecta I can change the scale but not have to redraw the picture, one picture works for a range of projectile masses and ejecta volume?

 

[pmacias]

I can say that there is indeed a simple formula for the ratio of projectile to ejecta mass. This formula is known as the conservation of momentum, which states that the total momentum of a system remains constant before and after a collision. In this case, the total momentum of the system would be the mass of the projectile multiplied by its velocity, which is equal to the mass of the ejecta multiplied by its velocity.

Therefore, the formula for the ratio of projectile to ejecta mass would be:

(Projectile mass x Projectile velocity) / (Ejecta mass x Ejecta velocity)

This formula would apply regardless of whether the projectile hits dust, rock, or a combination of the two. However, the mass of the ejecta may vary depending on the composition of the surface it is hitting. For example, if the projectile hits a rocky surface, the ejected material may contain larger and denser fragments compared to hitting a dusty surface.

Additionally, the size of the projectile may also affect the ratio of projectile to ejecta mass. A larger projectile would have a greater impact force and may result in a higher ratio of ejecta mass compared to a smaller projectile.

I hope this helps answer your question and provides some insight into the relationship between projectile and ejecta mass during a collision on the surface of the moon.