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?
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.
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