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I'm doing a design lab, where I have to "investigate the nature of craters formed on sand." I've done some research and figured out that the cube of the diameter ($d$) of the crater formed by an object is proportional to Kinetic Energy, i.e. $d^3$ is proportional to $\frac12mv^2$ (I got this information from this website, under the "Crater Size (Level Two)" heading).

I was wondering what would have to be added to this relationship (adding/subtracting numbers or adding in coefficients to either side) to turn this relationship into an equation. If all I have is the relationship and then I find a value for $m$ using this relationship and not any equation, will this value mean anything?

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Obviously the best way to determine this equation is to test it yourself!

Get yourself a notebook & pen, a heavy ball (like an 8 lbs bowling ball?) and a ladder and drive off to the nearest lake, beach, or sand pit. Drop the ball onto the sand from different heights (I'd say somewhere on the order of 2 dozen measurements will be a good enough supply for a good fit) and use the conservation of energy to say that the potential energy at the point of drop is equal to the kinetic energy at the bottom ($mgh=\frac12mv^2$).

Since you expect a linear relationship between $d^3$ and $\frac12mv^2(=mgh)$, $$y=Mx+b\longrightarrow d^3=M\left(\frac12mv^2\right)+b$$ you can use MS Excel to plot the two variables. From this data, you can fit a linear trend-line and get your exact equation! Note that if you don't drop the ball ($mv^2=0$) then there is no resulting crater ($d=0$), so you can expect that $b=0$.

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To say that "$d^3$ is proportional to $\frac{1}{2} mv^2$" is almost an equation. What you don't know is whether $d^3$ is half of $\frac{1}{2} mv^2$, or twice, or 10 times, right? This is called the constant of proportionality, and at this point you don't know what it is, so let's call it $K$. Now you can turn your relationship into an equation:

$$ d^3 =\frac{1}{2} K mv^2 $$

Once you know $K$, you can solve for any variable if you know the others. Your lab tests should give you an idea of the value of $K$, at least for the conditions you will be testing. You ought to be aware that, for real-world craters caused by actual astronomical bodies hitting the Earth, the exact mechanism is much more complex than what you are doing, so the actual crater sizes are somewhat different from what you will conclude. Still, it's a good first cut at understanding the process.

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  • $\begingroup$ hint: See if you can use math formatting in your posts for better readability. $\endgroup$ Commented Dec 2, 2014 at 14:29

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