Leia esta digressão em português.

A few months ago I watched this Veritassium video where Derek went around asking people to place where the Moon (tennis ball) is in relation to the Earth (basket ball) scaling for the size of the balls.

Not so surprisingly, everybody (besides one guy, I’d say) placed it much, much closer than it actually is and Derek gave a brief explanation of why people’s estimation of this distance is normally off and showed what would be the most correct answer.

It was only a few days later that it came back to me and I started thinking:

“Would I make a correct estimation? What if the Earth was of a different size?

And more importantly, how far the Moon actually is from the Earth?”

Well, thinking about all these, it came to me that something interesting to do would be trying to figure the ratio between the distance Moon-Earth and the radius, diameter or the circumference of the Earth. If we know this ratio, we can answer Derek’s question with any round object representing the Earth. But how can we know this ratio? How can we calculate it?

Surely the easiest method would be asking the internet. Pick our smartphone and ask Google, Yahoo or whatever other site you like. But that is the boring method. And not so impressive either. It’s much more impressive if we could do it using our own brain and the information within it. And here’s how we can make an estimation using just a few facts and the ability to use the power of 10:

First, let’s remind which information we need. We need the distance between the Earth and the Moon and at least the radius of the Earth (with it we can calculate both the diameter and the circumference).

Second, keep in mind that this is an estimation. We’ll round values up and down, focusing on the order of magnitude, not on the exact value.

So let’s get started!

To estimate the distance between the Earth and the Moon, we use the fact that light can do approximately 20 round trips to the Moon in one minute. This means that light takes 1,5 seconds to go from Earth to the Moon (60 seconds to go 40 times). And how fast does light travels? Well, we all know that light’s speed is approximately 3×10^8 m/s so in 1,5 seconds, light travels 4.5×10^8 m. Keep this number stored (and don’t forget the dimension!).

Now we need to find out the size of the Earth. There’s another video where we learn that, on average, a person walks the equivalent of 3 times around the world in a life time. Right, but is this someone from a rich or a poor country? Well, we don’t know, but we can consider that the average life time of the world’s population is about 80 years. We also know that the average person spends 1/3 of his/her life sleeping. 1/3 of 80 is about 30, so we have that this person is awake for 50 years. But wait, this doesn’t mean that a person walks whenever he/she is awake! We have to consider that during the younger ages we walk more but smaller distances (because our steps are shorter) and as we grow, although we start walking longer distances, we have less time to walk because we have to spend hours and hours sitting at school and, after that stage, we have to spend another handful of hours sitting at an office. So how long do we spend sitting around instead of walking?

Well, to estimate this, let’s consider we spend 8 hours sleeping and 8 hours at work/school/studying, leaving 8 hours for the other activities. “Other activities” includes commuting to work/school, taking care of daily needs besides sleep (eating, nature’s call, hygiene) and leisure. If we sum all the small walks inside the house, at the office and other commutes like going to the bakery down the street and consider that we all walk at least one hour a day to keep a healthy life, we can estimate that we walk around 2 hours a day.

So, 2 hours a day for one year means we walk for (365×2)÷24. We may not have a calculator at hand so let’s try to make this a little easier. We know that 365×2 is 730. But we also know that 24×30 is 720 so let’s approximate 365×2 to 720 and say that we walk for 30 days in a year. The average person walks for 30 days a year for 50 years, totaling 1500 days. This is equivalent to (1500÷365) years or approximately 1500÷360=150÷36=75÷18=37.5÷9. Well, 9×4 is 36 so let’s assume 1500÷365 is roughly 4 (for the record, it’s 4.109…).

Now we have the distance (three trips around the world) and the time (4 years), but we don’t have the speed! Well, based on my experience, on average, a person walks at around 5km/h. But hold on! The time we have is in **years** while the speed we have is in kilometer per **hour**!

No problem, we can find out how many hour there are in 4 years! 24 hours per day, 365 days per year. 365 times 24 is how much?

Again, we may not have a calculator at hand so let’s simplify and say that 24×365 is almost 20×370 (remember, we want to keep the order of magnitude, not the exact number!). It’s not that hard to multiply 20 times 370 because it’s the same as 37×2×100 which is 7400 (just for the record, 365×24 is 8760, which has the same order of magnitude of 7400, which is X×10^3 with X between 1 and 9.9).

So we now know that one year has about 7400 hours so in 4 years we will have about 29600 hours. Walking for 29600 hours at 5km/h we’ll travel around 150000 km (which is 30000×5), giving us a rough estimation of three times the circumference of the Earth. Divide it by 3 and we get 50000km.

Great! Now we have the circumference of the Earth in kilometers (5×10^4km) and we have the distance to the Moon in meters (4.5×10^8 m)! Let’s convert the distance to km and get 4.5×10^5km.

Now it’s time to find out the ratio between the distance and the circumference! So let’s just divide them!

(4.5×10^5)÷(5×10^4)=45÷5=9

Thus we found that we need to place the object representing the Moon at a distance of 9 times the circumference of the object representing the Earth.

And how accurate is this estimation we just did? Let’s use the internet to find out the values and compare!

First of all, the distance between the Earth and the Moon varies so we’ll use the average. According to Google, the distance is 384,400 km or 3.84×10^5km. We estimated 4.5×10^5km! We got the order of magnitude right! Way to go!

We also know that Earth is not a perfect sphere. But since this is just an approximation we can consider ignore that. According to the NGA, Earth’s circumference is 24,907 miles, which is equivalent to 40083.9 km or 4×10^4 km. We estimated 5×10^4! Again we got the order of magnitude right!

And what’s the ratio using these data? It’s (3.84×10^5)÷(4×10^4)=9.6

Close enough! We got the same order of magnitude for the ratio too!

If you want to know the ratio between the distance and the radius, we just need to take the circumference we found and find the radius. We know that the circumference is 2πr, where π is 3.14 and r is the radius. So, considering the circumference, the radius will be r=C÷(2π). We have C=5×10^4 so we divide it by 6, we’re left with r=0.8×10^4 km. Calculating the ratio using this value we get:

(4.5×10^5)÷(0.8×10^4)=56

Using literature’s value, we find out that the actual ratio is around 60!

Good enough, right?

And how and why could we get so close using so many assumptions and rough calculations? Simple!

The sum of all the rounding ups and downs compensates each other, leaving us with a result that is close enough and within the same order of magnitude! But you need to keep your estimations reasonable so don’t stray away from reality! One place that we could make a completely wrong estimation is how for long a person walks. If we considered 10 years, the final result would be in a different order of magnitude and the final ratio would be way off! But we would still get a way better answer than this image: