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Suppose a car is traveling 5 m/sec. So after 3s it has traveled (3s)*(5m/s) = 15m. Of if the car has traveled 10m then it took (10m)/(5m/s) = 2s.
Suppose you used the conversion factor incorrectly, i.e. used the inverse. So for the latter example, after 10m we have
$$ 10m * 5m/s = 50 \, m^2/s $$ Does this have a reasonable interpretation?

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  • $\begingroup$ is there a quantity with these units? $\endgroup$ Jan 12, 2022 at 1:28

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This would not have any reasonable interpretation within the context of the problem. It may happen to be the correct answer to some other problem involving impulses and accelerations, but it is simply not a correct way to go about the problem.

Another way to think of it is that we can say "speed is distance divided by time," but this does not mean there is any particular relationship between "speed" and "distance multiplied by time."

This can be seen most obviously if we say $x = \frac y z$ and pick $x=1$ as our particular example. This means $1=\frac y z$, but it is trivial to see that we know little about $yz$. We do know that $yz=yy=y^2$, but that shows that $yz$ could quite literally be any positive value. The division of two values is unrelated to the product of those two values.

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