How high up is outer space?

The Kármán line is the altitude of the boundary between earth’s atmosphere and outer space. This $100$  km or $328, 084$ ft. The value comes from Fédération Aéronautique Internationale, and it’s the same value that NASA uses to define the boundary between our planet’s atmosphere and outer space. 

If you’re like me, the highest you've ever been from sea level is around $30,000$ ft to $40,000$ ft, which is the range of altitudes at which most commercial airliners cruise at. 
For some context, the tallest mountain on earth is Mt. Everest, with a peak at $29,029$ ft, measured with respect to sea level.

The boundary between earth and outer space, at $328,084$ ft, is roughly 11 times higher than Mt. Everest, as well as the highest up you’ve probably ever been. Try to picture that for a  moment.

If this picture doesn’t give you some sense of awe, perhaps another way of thinking about it will. Let's think about how much kinetic energy is gained due to the force of gravity at this altitude. After all, if you were to jump from such a height, the gravitational field of earth would impart onto you energy in the form of motion. Gravity would accelerate you to some speed. So, a natural question that arises is, if you were to fall back to earth from the Karman line, how fast would you be moving when you hit the ground? We'll calculate an upper bound on this speed. That is, what is the fastest you would be moving when you hit the ground if you experienced zero air resistance. 

The acceleration of gravity on earth is $g = 9.8 \frac{meters}{seconds^2}$ or $ 21.9\frac{miles}{hour^2}$.

Using one of the kinematic equations, its possible to determine the speed at which you will hit the ground.

$$d = vt + \frac{1}{2}at^2$$

In case you want to work it out yourself, $d$ is distance, $t$ time, $a$ acceleration (in this case $g$).

First, you must solve for the time it takes to fall a distance of $328,084$ ft. It would take you $143$ seconds or about $2.5$ minutes. For each hour, youd accelerate by $21.9\frac{miles}{hour}$ because of gravity.

By the time you hit the ground, youd be moving at a speed of $3,131 \frac{miles}{hour}$.

Again, keep in mind this back-of-the-envelope calculation ignores the effect of drag -- air resistance -- experienced by a falling person. So, you will actually be moving much slower than this, but this gives you some sense of the amount of energy gained from gravity at this altitude. 

So, how high up is outer space? High enough up that if you jumped from there, and didn't experience  air resistance, youd reach a speed of $3,131 \frac{miles}{hour}$. Thats faster than the speed of sound ($767 \frac{miles}{hour}$). If you kept moving at that speed, you could travel from Los Angeles to New York in less than an hour. 

Of course, once you factor in air resistance, you'll find that you reach a terminal velocity. This is a calculation best left for another essay, but it's around half the speed predicted by the model of no air resistance. For some context, competitive skydiver Felix Baumgartner holds the record for terminal velocity reached via skydiving. He reached a velocity of $834 \frac{miles}{hour}$ by jumping from 128,100 ft, about 40% of the altitude of the Karman line.

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