Is there an equation I can use to determine the volume of a crater in my 'drop the ball on sand' experiment? I looked at the equations for volume of hemisphere and cone but do not seem to fit the shape of the crater. The crater looks like a cone but has a spherical cap instead of a sharp point which gives it a bowl-shape.I have measurements of the depth and diameter of the crater but that's about it. I am hoping to find he relationship between the energy of the impactor and the volume of the crater
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1$\begingroup$ Related questions by OP: physics.stackexchange.com/q/195336/2451 and physics.stackexchange.com/q/195152/2451 $\endgroup$– Qmechanic ♦Commented Jul 25, 2015 at 7:17
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$\begingroup$ I think you'll need to tell us a bit more about this experiment in order to make this a reasonable question - at least, what information do you have access to that you could use to determine the volume of the crater? $\endgroup$– David ZCommented Jul 25, 2015 at 13:17
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$\begingroup$ I've edited the question $\endgroup$– Luqman HalimCommented Jul 26, 2015 at 0:41
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$\begingroup$ Treat the crater as a truncated cone (volume = difference of volume of "big, complete" cone minus "small, missing bit" of cone, plus a slice from a sphere). I have a drawing but until the question is re-opened I can't post it. $\endgroup$– FlorisCommented Jul 26, 2015 at 2:44
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2 Answers
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If you draw the following diagram:
you can see that the volume of the crater is the volume of the "truncated inverted cone" plus the volume of the bit of sphere. Since the volume of a cone is $V=\frac13 A h$ where $A$ is the area of the base and $h$ is the height, the volume of the truncated cone is given by
$$V_{cone} = \frac{\pi}{12}(d^2(h+h')-d'^2(h'))$$
The volume of the salmon-colored bit of sphere can be found by integrating. Luckily, Wolfram already did the hard work and we can start with their result:
$$V_{cap} = \frac{\pi}{3}c^2(3R-c)$$
Actually they used $h$ for the height of the cap, but we already use that for a different quantity - so I will define the height of the cap as $c$.
In my drawing, we can deduce the height of the cap from the angles $\alpha$ and the fact that $\beta = \pi - \alpha$ so $\cos\beta = -\cos\alpha$:
$$c + r\cos\beta = r\\ c = r(1+\cos\alpha)$$
We can also solve for $d'$ in terms of the other quantities. I will leave it up to you to take it from here.
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$\begingroup$ Vcap=π3c2(3R−c). In this equation, is the R represented similar to r in the picture? $\endgroup$ Commented Aug 5, 2015 at 11:53
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$\begingroup$ Sorry yes capital and lower case r intended to be the same $\endgroup$– FlorisCommented Aug 5, 2015 at 14:23
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$\begingroup$ I have decided to calculate the volume of crater as a spherical cap. Also, in the spherical cap equation, is c^2 proportional to the volume? $\endgroup$ Commented Oct 7, 2015 at 9:36
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You could use two approaches. Sounds like now you're trying to carefully characterize and measure the shape of the crater, and then calculate the displayed volume. It sounds like the shape isn't completely regular, though, so you may have a tough time doing this accurately.
Another approach would be to empirically measure the displaced volume of sand. Let's say your sand box is a wooden frame with a bottom holding sand. Before the trial you fill the box with sand and then use another board to scrape off any excess, leaving the surface level with the edges of the box. The impact will spray sand out of the crater onto the rest of the surface. So, you then scrape this build-up of sand off, and measure its volume.
This presumes that the sand doesn't compact, which probably isn't accurate. But then again, that may be the same assumption that measuring the size of the hole makes.
answered Jul 25, 2015 at 12:41