While there is no preferred reference frame for motion, rotation is straightforward. A reference frame that experiences no pseudo forces such as centrifugal force is not rotating. A better practical reference is to simply look at distant stars, for all purposes they appear stationary.

So because you have a handle on rotations, its very straightforward to count that it takes a day for Earth to do a trip around it’s own axis of rotation and a year for Earth to do a full trip around the Sun. Knowing the orbital radius of Earth is 1AU, it’s easy to calculate that Earth’s linear speed around Sun is pi*2 AU/year or about 30km/s

For the spin, we have the Sun. That isn’t truly stationary but it’s close enough; pound a stake into the ground, wait for its shadow to get to its shortest, start a stopwatch, the next day wait for the shadow to get to its shortest, stop the stopwatch and you’ve measured how fast the planet spins. (That’s the synodic day which is slightly different from the sidereal day, but you’re five so let’s keep it simple. :D)

For the orbit – imagine you’re looking at a billboard, and then you take six steps to the left and look at it again. It’s not a big difference, but you’re seeing the billboard from a slightly different angle now and if you took some careful measurements, you could tell that you’d moved. Conversely, if you know how far you’ve moved and take some measurements on the billboard, you could determine how far away the billboard is. A Greek philosopher whose name isn’t coming to me took some readings of the Moon’s location, and arranged for a colleague in another city to take readings on the same nights, and then they did some geometry and figured out how far away the Moon was. (They did damned well – less than 10% off the figure we believe today.)

Once they knew how far the Moon is…they figured out that when we’re seeing a half-moon, that must mean that Earth, the Moon and the Sun are at a 90-degree angle. If you know the 90, the angle from Earth to the Sun, and the distance from Earth to the Moon, you have enough to totally solve that triangle and now you know how far Earth is from the Sun. Multiply that by 2pi and you know the circumference of our orbit; divide that distance by 365.25 days and you have our orbital velocity.

We know that our distance to the sun is 149.6 million kilometers.

From that we can calculate the circumference of its orbit, which is about 940 million kilometers.

That divided by the time it takes the earth to make one full round gives us a speed of 107,200 km/h or ~30 km/s.

Which is still relatively slow, considering that our entire solar system is orbiting the center of the galaxy at the mindboggling speed of 720,000 km/h.

Hold on to your butts!

In addition to the other answers, you can also measure the speed of the earth in relation to the cosmic microwave background, which is the light leftover from the early universe. The CMB is mostly the same everywhere, so it’s a pretty good point of reference for how fast a given object is moving through space.

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