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I'm having trouble solving this Physics question with "made up" units. Here is the problem and my work so far.

The Omicronians have told you that they use the units: $$\\{OP=7.29\frac{flurg}{grom^2}}$$ These units need to be converted to meters/second^2 and their respective values are:

5.24 flurg = 1 meter

1 grom = 0.493 second.

Question: What is the value of OP in meters/second^2?

Below is one of my many failed attempts at solving this. Do I actually square the value of "grom" or leave it in the same format as m/s^2? Either way, my answer was still incorrect when I tried to square the denominator, leading me to believe I've gone wrong elsewhere.

$$\\{7.29\frac{5.24m}{0.493s^2}} = 77.48m/s^2$$

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  • $\begingroup$ i can attach a screenshot of the problem if that's allowed? $\endgroup$ – JCD133 Jan 26 '18 at 3:32
  • $\begingroup$ Look at it this way. If you measured a distance of 5.24 flurg is the number of metres larger or smaller than 5.24? So if a measurement was 5.24 flurg/s$^2$ is the number of m/s$^2$ larger or smaller than 5.24? $\endgroup$ – Farcher Jan 26 '18 at 3:40
  • $\begingroup$ meters are longer than flurgs, so the number of meters is smaller. The second part of your question is where I'm losing any sense of what's happening. in m/s^2, I'm not usually squaring the denominator, it's just the unit of measurement for acceleration. And, if I do square it.... value is 1 regardless. In contrast, 0.493 squared isn't intuitive for me at all $\endgroup$ – JCD133 Jan 26 '18 at 3:46
  • $\begingroup$ Think of area (instead of time) and the conversion from one unit of area to another. Then think of the conversion of reciprocal area. $\endgroup$ – Farcher Jan 26 '18 at 4:25
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$$\require{cancel}$$

The trick to unit conversions is to think of it as multiplying by 1. When you have $$5.2\ flurg = 1\ meter,$$ that also means that $$\frac{5.24\ flurg}{1\ meter} = 1$$ and $$\frac{1\ meter}{5.24\ flurg} = 1.$$ This is important because converting units does not change the quantity, only the paper-and-pencil representation. Which ratio you use depends on how you want the units to cancel (which works just like canceling numbers in fractions). So, for every unit you want to convert, set up a ratio like one of the above and multiply it by the original quantity to get a new quantity in new units.

For example, $1\ foot = 12\ inch$. So, to convert 1.5 feet to inches $$1.5\ feet \underbrace{\left(\frac{12\ inch}{1\ foot}\right)}_{=\ 1} = 1.5\ \cancel{feet} \left(\frac{12\ inch}{1 \cancel{foot}}\right) = 18\ inch$$ Notice that using the other ratio leads to a useless answer: $$1.5\ feet \underbrace{\left(\frac{1\ foot}{12\ inch}\right)}_{\textrm{also 1, but not helpful}} = 0.125\ \frac{feet^2}{inch}$$

For units with powers, use the same power on the conversion ratio. $$9.8\ \frac{m}{s^2}\left(\frac{60\ s}{1\ min}\right)^2 = 3.5\cdot10^4\ \frac{m}{min^2}$$

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