Talk:Acceleration due to gravity

There is a problem with answer at bottom of altitude section...

Is multiplying 9.8(Earth's grav. const., in m/s^2) by 6371 km, not m, which would be unitarially correct. Should be, upon converting to proper units, the correct equation:

(9.8 * ((6371 * 100)/3600)^2)/((7371 * 100)/3600)^2 = 7.32... — Preceding unsigned comment added by 2602:306:ceb6:95d0:6918:2973:36f7:59aa (talk • contribs) 02:59, 21 September 2018 (UTC)

Let us look at the equation with values and units,

${\frac {9.8{\text{ m}}{\text{ s}}^{-2}\times (6371{\text{ km}})^{2}}{(7371{\text{ km}})^{2}}}\ .$

Now let us only think about the units,

${\frac {{\text{m}}{\text{ s}}^{-2}\times ({\text{km}})^{2}}{({\text{km}})^{2}}}\ .$

We have $({\text{km}})^{2}$ on both the top and the bottom of the equation, so the units cancel down to

${\frac {{\text{m}}{\text{ s}}^{-2}\times {\cancel {({\text{km}})^{2}}}}{\cancel {({\text{km}})^{2}}}}\ ={\text{m}}{\text{ s}}^{-2},$

which are the desired units of acceleration, as expected.

Whatever units the squared lengths on the top and bottom of the equation are measured in, as long as they are the same units, they cancel out.

For example, if we use metres instead of kilometres,

${\frac {{\text{m}}{\text{ s}}^{-2}\times {\cancel {({\text{m}})^{2}}}}{\cancel {({\text{m}})^{2}}}}\ ={\text{m}}{\text{ s}}^{-2}.$

Hence, we get the same value, 7.3, as the method in the article. (The answer, 7.3 m/s², has 2 significant figures because the acceleration due to gravity, 9.8 m/s², is only quoted with 2 significant figures.) What we have here is a ratio of the squares of two lengths (or, equivalently, the square of the ratio of two lengths), which is a dimensionless number because the lengths are measured with the same unit.

To convert kilometres to metres, the kilometre values need to be multiplied by 1,000 instead of 100, as you have done. There is no need to divide by 3600. You get the same answer anyway because you have done the same to the numerator and the denominator.

In most cases, it is best to convert units to base SI units first of all, as you say. The article uses kilometres there instead of metres because, in this instance, it seems more convenient since the units cancel out anyway. We could convert kilometres to metres so that everything is in base units, but we do not need to.

We could do a conversion of km to m in the equation, as you have done, or do a step-by-step conversion, but I think this might complicate things and distract the reader from the main line of thought, especially if he or she is not familiar with unit conversion. We could quote the values in full, e.g., 6371000 m, but this means the values are long, and also potentially confusing because it is unclear that the three zeros are not significant figures. (We do not want people to see the zeros and, since there is no note to the contrary, assume that the zeros are significant. And I guess we do not want to introduce the idea of significant figures in case readers are not familiar with it.) We could use scientific notation and say, e.g., 6.371×10⁶ m, but this means that the reader has to understand scientific notation.

On the other hand, we are not using base units, which is not a good idea in general. It might be a bad example and encourage readers to be careless about units, and readers may not understand the unit cancellation described above. We could rework the derivation somehow so we get something like

${\text{gravitational acceleration at altitude}}={\text{acceleration due to gravity at surface}}\times ({\frac {\text{distance from centre to surface}}{\text{distance from centre to altitude}}})^{2}.$

This might make the unit cancellation clearer to readers, but I'm not sure if this would be the best approach.

Overall, I am conflicted about this, but at the moment it seems to me better to use km. We could add a note to say that the units for the lengths on the top and bottom of the equation need to be the same so they cancel out, as another suggestion, for example. The whole section may not be very clear and may need a rewrite. I welcome the thoughts of others. --Thrasymedes (talk) 00:41, 5 February 2020 (UTC)

{{subst:Requested move|Standard gravity |reason= Simplified, more common title.}}