I have read different speeds of Earth in different sources. $382\;{\rm km}/{\rm s}$, $12\;{\rm m}/{\rm s}$ and even $108,000\;{\rm km}/{\rm h}$. Basically, it's moving too fast around the Sun. And the Solar System is moving too. So why don't we feel it and why doesn't it harm us in any way? Inertia can only be a part of it. But what's the whole reason?
Newtonian Mechanics – Why Earth’s Speed Through Space Does Not Affect Us
newtonian-mechanicsreference framesrelative-motion
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The atmosphere rotates along with the Earth for the same reason you do.
Force isn't needed to make something go. That's a basic law of physics - that a thing that's moving will just keep moving if there's no force on it.
Force is needed either to make something change its speed, or to make its motion point in a new direction. A force can do both or just one of these. Most forces do both, but a force that pushes in the exactly the same direction you're already going only changes your speed, and does not change your direction. A force that pushes at a right angle to the direction you're already going only changes your direction, and does not add any speed. A force at "10 o'clock", for example, will change both your speed and your direction.
As you stand still on Earth, you continue going the same speed, but your direction changes; between day and night you move opposite directions. So the forces on you must be at a right angle to your direction of motion. Indeed, they are. Your motion is from west to east along the surface of the Earth, and the force of gravity pulls you down towards the center of the Earth - the force and your motion are at right angles. Similarly for the atmosphere. It is moving along with the Earth, and moving at a constant speed. It does not need anything to push it along with the Earth. Since only its direction of motion is changing, it only needs a force at a right angle to its motion, the same as you, and the force that does the job is again gravity.
That's not the whole picture, because the amount that your direction of motion changes depends on how strong the right-angle force is. It turns out gravity is much too strong for how much our direction of motion changes as the Earth spins. There must be some other force on us and on the atmosphere canceling out most of the gravity. There is. For me it's the force of the chair on my butt. For the atmosphere, it's the air pressure.
So gravity doesn't "make the air rotate". The air is already going, and gravity simply changes its direction to pull it in a circle.
You may be wondering why the air doesn't just sit there and have the Earth spin underneath it. One answer to that is that from our point of view that would mean incredibly strong wind all the time. That wind would run into stuff and eventually get slowed down to zero (that's from our point of view - the air would "speed up" to our speed of rotation from a point of view out in space watching everything happen). Even the air high up would eventually rotate with the Earth because although it can't slam into mountains or buildings and get stopped from blowing, it can essentially "slam into" the air beneath it due to friction in the air. (This is a little redundant with dmckee's answer; I was half way done when he beat me to the punch)
Velocity does indeed have to be measured relative to something. We can measure our radial velocity relative to any other astronomical object we care to, by measuring Doppler shifts. But if you want to know our velocity "relative to the Universe as a whole" rather than relative to any one object, we have to be a bit careful to define our terms.
Because the Universe appears to be approximately homogeneous and isotropic, it makes sense to define a "rest frame" at any given point. (The rest frames at different points are moving with respect to each other -- that's what it means to say that the Universe is expanding.) This "rest frame" is essentially the frame in which the stuff surrounding that point appears to be moving isotropically (the same in all directions). In practice, the best way to define that rest frame is to find the frame in which the cosmic microwave background appears the same in all directions (has no dipole moment, to be precise). Relative to this frame, the local group of galaxies is moving at about 600 km/s (Wikipedia gives precise numbers and probably citations that I'm too lazy to look up).
People sometimes worry about whether the existence of a preferred "rest frame" of this sort is in conflict with the principle of relativity. The answer is that it isn't. There are a couple of ways to see why. One is to note that the principle of relativity says that the laws of physics have no preferred frames, but particular solutions to the laws can have preferred frames. Another way of putting it, which I prefer, is that the "rest frame" we use in cosmology is simply the center-of-momentum frame of a bunch of particles (namely the CMB photons in our neighborhood). In other contexts, we're not surprised or worried by the fact that a bunch of particles have a rest frame, so why should we worry about it here?
Best Answer
Speed doesn't kill us, but acceleration does.
When astronauts go into space at launch and when fighter pilots turn very tight turns at high speed they experience 'high g forces' - their bodies are accelerated very fast as they accelerate and gain speed to go into space or as the direction of their speed changes. One of the problems with this is that for fighter pilots the blood can rush to the feet (black out) or to the head (red out). Too much acceleration makes people pass out and could at extremes be fatal I guess.
To go around the sun in (nearly) a circular path we are acclerated by the gravity from the sun. The acceleration can be calculated by $v^2/r$ where $v$ is our speed and $r$ is the distance to the centre of the sun. This acceleration turns out to be $\sim~0.006~m/s^2$. By contrast the acceleration that we feel here at the surface due to the gravitational pull of the earth on us is $\sim~10~m/s^2$. So the acceleration due to travelling around the sun is so small we don't notice it. We do notice the pull of gratvity from the earth on us, but our bodies are used to it and can cope with it.
To think about it another way we can go very fast in a car on a motorway/highway without noticing it, the big danger is having to stop very quickly or crashing when we change speeds very rapidly - acceleration is the rate of change of speed so changing speed very rapidly is equivalent to a very high acceleration - in a car we might call this deceleration.
[for calculation above $v=3 \times 10^4~m/s$ and $r=1.5 \times 10^{11}m$]
after good comment from hdhoundt - For astronauts in orbit (e.g. in the space station) they can cope with the acceleration they experience, which holds them in orbit around the earth. Indeed they feel weightless because they are not held by the gravity of earth on the surface. Instead they and their surroundings are in 'constant free fall'. The speed of the space station in orbit is $7.71 km/s$, which is $\sim~ 17,000 ~mph$.
Full discussion of this topic might venture into relativity, but I think that is beyond the scope of the question.
after good comment from Mooing Duck -
Perhaps even more dangerous than acceleration is jerk, which is the rate of change of acceleration and other higher order terms. Jerk would be very severe in the case of car collisions. - But also if the driver of a car or bus has to 'brake' and slow down very suddenly it can be very uncomfortable for the passengers.
After good comment from Jim (and Cory)-
Good point raised about acceleration and/or jerk on a human body. If every part (and every particle) of the body experience the same acceleration or jerk then the body will suffer significantly less (possibly no) damage compared to when one part of the body is accelerated of jerked and the acceleration or jerk is transmitted to other parts of the body by the structure of the body. The classic example here is 'whiplash' neck injury, where a jerk on the body is transmitted to the head through the neck. To reduce the damage this may cause seats in cars generally hare head rests that will support the back of the head and for people who are involved in motor sports (e.g. car racing) may wear a neck brace/support that prevents the head from swinging backwards and forwards on the neck in the event of a collision.
Another aspect of acceleration to all part of the body concerns rocket launch for astronauts. The rockets will be designed so that as much as possible all part of the body are equally supported and the body lies 'flat with respect to the acceleration' so that the blood in the astronaut's body does not rush to the feet or head. This is a serious consideration and Memory Foam came from research by NASA into safety for aircraft cushions and helped cushion astronauts in rockets.