3550.23 – Our Whirling, Twirling, Swirling World

We humans are constantly traveling around in circles while rarely ever thinking about it. We get a free ride all the way around Earth’s axis every day, and another one around the sun every year!

The diameter of the Earth is approximately 12,756 km at the equator. Assuming that the Earth is perfectly spherical (it's not) and that there are exactly 24 hours in a day, how fast are you traveling if you are standing in the same spot on the equator, ignoring the motion of the Earth around the sun?

Additionally, keeping in mind that the distance from the sun to the Earth is approximately 149,600,000 km, and there are about 365.26 days in a year, how fast is the Earth traveling at a given instance, assuming that it takes a perfectly circular path around the sun? Give all of your answers in kilometers per hour.

Solution

This is a rather wordy problem. You probably noticed that it is really two problems rolled into one.

Considering the first problem, we can easily calculate the circumference of the Earth by multiplying its diameter by π\pi :

π×12,756 km≈40,074 km\pi \times 12,756 \text{ km} \approx 40,074 \text{ km}

Because we are ignoring the Earth’s movement around the sun for this problem, we only have to find out how fast a point is moving if it travels 40,074km40,074 km in 24hours24 hours. This can easily be calculated:

40,074 km24 hours≈1,670 kmhour\dfrac{40,074 \text{ km}}{24 \text{ hours}} \approx \dfrac{1,670 \text{ km}}{\text{hour}}

The second problem is essentially the same. First, we find the distance the Earth travels around the sun:

2π×149,600,000 km≈939,964,522 km2\pi \times 149,600,000 \text{ km} \approx 939,964,522 \text{ km}

Remember to use 2π2\pi , because we are given the radius rather than the diameter of the circle traced by Earth as it moves around the sun. We now know how far the Earth travels, and we know that the time taken to complete this journey is 365.26×24 hours≈8,766 hours365.26 \times 24 \text{ hours} \approx 8,766 \text{ hours}. We can now divide, as before, and arrive at our answer:

939,964,522 km8,766 hours≈107,228 kmhour\dfrac{939,964,522 \text{ km}}{8,766 \text{ hours}}\approx \dfrac{107,228 \text{ km}}{\text{hour}}

(OK: to convert km to miles, multiply by 0.6213699495.) (Actually, that last result is interesting: it's about 1000 times as fast as we drive on the highway.)

By the way, the earth isn't a perfect sphere--it bulges out around the equator. It also has tall mountains and deep chasms as well. But if you ask students what is the most perfect sphere on the planet--thinking of smooth billiard balls, etc., the answer is that the most perfect sphere on the planet is the planet itself.