A Mountain of Myths about Space Launch
Altitude 
Density 
Year 
Record 
km 
atmos. 


0.0 
1.00 

Standard density 1.225 kg/m³, varies 
8.848 
4.07e1 
1953 
Everest, Highest Mountain 
37.65 
4.62e3 
1977 
Highest Airbreathing Manned Aircraft 
41.4 
2.66e3 
2014 

53.0 
5.86e4 
2002 

99.0 
5.49e7 

"in" the atmosphere, national airspace 
100.0 
4.58e7 

Legal Boundary of Space 
101.0 
3.83e7 

"out of" the atmosphere, international 
160.0 
1.01e9 

rapidly decaying single orbit 
200.0 
2.1e10 

practical short term orbit  days? Depends on area and density 
400.0 
2.3e12 

International Space Station  orbit decays 2 kilometers per month 
Space is high vacuum. Zero lift for wings or balloons. The legal boundary of 100 km (the Kármán line ) is convenient but entirely arbitrary; it is twice as high as any atmospheric vehicle can fly or float, and half as high as any practical satellite can orbit.
Tourist suborbitals to 100 km ( such as SpaceShipOne ) are not practical spacecraft, merely dangerous and ostentatious entertainment, somewhat like climbing Mount Everest but without the strenuous exercise or rigorous accomplishment. They will measure your money, not your mettle  if one of these projects ever achieves FAA certification.
Orbits: Velocity, not Altitude
Real space is orbit, and the lowest practical orbit is 200 km. Even that is far too low for a large, highdrag object like ISS, which requires frequent reboost to compensate for air drag, even though the air is 2 parts per trillion of the density at sea level. That is a difficult number to grasp  a volume of "ISS vacuum" as big as AT&T Stadium in Texas (104 million cubic feet, 3 million cubic meters) would fit in 7 cubic centimeters, half a tablespoon, if compressed to sea level density.
While circular orbits in the equatorial plane of the Earth are the easiest to think about, real orbits are usually ellipses, in planes inclined at an angle to the equator; that adds an additional four defining parameters to the orbit description. Additional parameters describe how the orbit decays over time, and how the angles and eccentricities change due to the Earth's equatorial bulge, and tidal forces from the Sun, Moon, Jupiter, and other planets. Just getting above the atmosphere is not nearly enough to get to a useful orbit, and changing orbits can be more propellantcostly and timeconsuming than getting to orbit in the first place.
Circular orbits balance centrifugal acceleration of travelling in a circle at high velocity, with the centripedal acceleration of gravity:
\Large { { v^2 } \over R } ~=~ g ~=~ { { G M } \over { R^2 } } ~=~{ { \mu } \over { R^2 } }
\mu ~=~ GM is the standard gravitational parameter; 398,600.4412 km³/s². We can measure \mu to 10 decimal places because of ultraprecise measurements of orbiting satellites, but we cannot estimate either G or M alone to six decimal place accuracy. For calculating orbits, this is irksome but not very important. A more important effect is J_2 , a small asymmetry in the Earth's gravity field caused by the equatorial bulge. We won't go into that complication here, but J_2 and other small effects are important for high precision or long duration orbital calculations.
The radius of the Earth is on average 6371 kilometers, 6378 kilometers at the bulge of equator. The International Space Station (ISS) is in a typical Low Earth Orbit. Slightly elliptical, ranging from 402 km to 409 km altitude, with an inclination of 51.6°. The ground track goes as far north as Calgary and London, and as far south as the Falklands and Tierra del Fuego. We can approximate that orbit very crudely as a 6780 kilometer circular orbit:
\Large v = \sqrt{ \mu \over R } = ~ 7668 m/s
Going to that orbit is aided (somewhat) by the rotational velocity of the Earth, and hindered by the additional 400 kilometer altitude climb. Launching to that orbit requires a horizontal velocity change of approximately 8000 meters per second.
In addition, because ISS is in an inclined orbit (51.64°), a launch to match that orbit must be in the northeast or southeast direction (launch azimuth). From Kennedy Space Center, launch azimuth is constrained to orbits with inclinations between 39° to the south and 57° to the north, so that expended first stages and failed launches do not drop hypervelocity debris on inhabited areas.
Note: The high inclination orbit of ISS allows easier access from the Baikonur Cosmodrome at 46° N in Kazakhstan, and Russia's new Vostochny Cosmodrome at 51.9° N in the Russian far east. This orbit also maximizes overflight opportunities over the United States, an advantage that compensates for the extra difficulty of reaching this orbit from KSC. However, reaching this orbit from a circular equatorial orbit requires a large and costly velocity change of 6.7 km/s (mostly "sideways"), so it is easier to reach the ISS orbit from a properly inclined ground launch than from an equatorial orbit.
The old slogan "orbit is halfway to anywhere" is mostly false; large orbit inclination changes (to polar or sun synchronous orbits, for example) can be more costly than ground launch. A direct launch to Pluto and beyond from low Earth orbit requires more than 14 km/s of additional ΔV. That additional velocity can be slowly added with low thrust, propellantthrifty electric rockets , and this is used for GEO satellites and the Dawn space probe to the asteroid belt, but the ion thrusters took over only after a traditional chemical rocket provided the escape velocity. I hope that someday we will have ion engines that can take us from Low Earth Orbit to the far planets, but not yet.
Launches from exotic locations (like mountaintops) would be similarly constrained; this rules out mountains in the Western hemisphere (no tall mountains on the east coast), and most mountains in the eastern hemisphere. Mountains in the sparsely populated Himalayas may be acceptably safe by Asian standards; the inhabitants tolerate much higher risks from earthquakes. However, China moved their main launch center from a higher inland location to the east coast of Hanian Island, so they are more protective of their populations than some Westerners might assume.
Altitude: Reduced Aerodynamic Drag, not Gravity
Yes, gravity goes down with altitude, but not very much. g = { { \mu } \over { R^2 } } = 9.80 m/s² at the surface, 8.67 m/s² at 400 km ISS altitudes, a 12% reduction. Aerodynamic drag drops by a factor of 400 billion. So, if ISS descends 2 km per month in the low drag atmosphere at 400 km altitude, it would descend 400 km in 1.2 milliseconds at ground level air pressures. Actually, it would smash into incandescent fragments in that time.
The space shuttle throttled back engine power to minimize aerodynamic stress. That reduced the needed structural strength and mass. At 60 seconds into the flight, the maximum aerodynamic pressure (maxQ) was 720 psi at 34,000 feet altitude and 1400 feet per second velocity, which is 5 MPa (50 atmospheres), 10.4 km, and 430 m/s. Air density is 32% of sea level at that altitude. If the shuttle was lifting at full throttle 3 gees (including gravity) 19.6 m/s² all the way from the pad, it would be ascending 640 m/s at that altitude, creating 2.2 times the aerodynamic pressure and structural load. That extra structural mass would require extra reentry protection mass, and that extra mass is subtracted from payload mass, which is the reason for this whole exercise. It is better to lift a little extra fuel for the first minute of the flight, than to vastly increase structural mass and reduce payload mass for the entire mission.
The aerodynamic pressure is proportional to the velocity squared. If for some crazy reason the shuttle was moving at full orbital velocity (about 8000 m/s) rather than 430 m/s at 10.4 km altitude, the structural load would increase by a ludicrous factor of 350. If this occured at Everesttop level (8.8 km) rather than 10.4 km, the air density would increase by 16%. and the structural load would increase by a factor of 400. The space shuttle would have to be solid metal to survive that.
The space shuttle accelerates faster after maxQ, as the engines are throttled up and fuel mass is reduced. At 120 seconds, it is 45 kilometers high, the air density is 0.15% of surface density, and the velocity is 1300 m/s, still only 16% of orbital velocity, but the aerodynamic pressure has dropped to 30 psi or 220 kPa. The nose heating, which is proportional to the density times the cube of the velocity time the density, drops to 13% of the nose heating at maxQ.

Pardon Ye Olde English units  that's the way the space shuttle was designed. 150,000 feet is 45 km, far below the 80 km altitude where the shuttle pitches over and accelerates to 7600 m/s or 25,000 feet per second 
At 80 km altitude, the shuttle is moving at full orbital velocity, in an elliptical transfer orbit to its destination in Low Earth Orbit. The velocity is approximately 7600 m/s at this altitude, relative to the air that is rotating along with the Earth. The density is 15 ppm of surface density, and the heating increases to 26% of maxQ heating, but diminishes as the shuttle's orbit ascends towards the destination orbit.
A launch loop will begin its transfer orbit at 100 km altitude, 0.46 ppm of surface density, so the heating will be 0.8% of shuttle maxQ. A launch loop climber vehicle will undergo more heating and aerodynamic drag as it ascends the acoustic tether elevator, but that will be limited to lowerthanshuttle values to reduce loading on the elevator tether. Unlike the shuttle, launch loop vehicles will not carry large wings, external tanks, and solid rocket boosters through maxQ, so the total aerodynamic load will be smaller.
A 400 kilometer drop from Space Station altitude, zero orbital velocity
If you were suddenly transported to 400 km altitude (in a space suit) and dropped from there, you would not orbit the Earth. You would fall down, your vertical speed increasing rapidly. After 260 seconds, you would plunge through the 100 km altitude Kármán line at 2330 meters per second, 8 times the speed of sound at that altitude, but you would notice nothing, a 0.3 Pascal pressure force added to the 100 kPa (100,000 Pascal air pressure in your suit.
Things get frisky during your final 40 second plunge deep into the atmosphere, approaching a terminal velocity of 2700 m/s. At 7.1 kilometers altitude (higher than all mountain peaks in the Western Hemisphere), the air density is about half of sea level. However, the wind pressure beneath you is 40 times air pressure, about 400 tonnes per square meter; you would be smashed to jelly, though you would not burn up like a (vastly faster) small meteor.
 Actually not, you slam into a wall of denser air at a much higher altitude. You will stop accelerating and slow down when the drag pressure equals gravity. If your effective drag area is 1 square meter and you and your suit weigh 100 kilograms, then 1000 Pascals of pressure will start slowing you down at an atmospheric density around 170 ppm of surface, around 60 kilometers up. Air density (and drag pressure) doubles every two seconds, so within 10 more seconds of downward plunge, your suit will be ripped off by turbulence, and you will be crushed by gee forces.
Meteors arrive at 15 to 30 thousand meters per second. Orbital velocity is around 8000 meters per second, and air drag force is proportional to velocity squared. So, entering at orbital velocity is about 9 times the drag force of a 400 kilometer plunge, and vastly more than encountered by SpaceShipOne, though 4 to 14 times lower than a meteor. Meteors (and reentering spacecraft) slow down higher in the atmosphere; the smallest meteorites may burn up, but most hit the ground at bullet speed, with a charred surface and a frozen (by deep space) center. As we tragically learned in the catastrophic entry of Columbia, damaged vehicles are ripped to shreds in the middle atmosphere (70 km for Columbia, air density 70 ppm of surface density), but most fragments impact the ground intact.
 Launch Loop (2017 design) vehicles reach orbital velocity at 100 km altitude, 0.5 ppm of surface density, so the drag pressure is 140 times lower than 70 km shuttle entry altitude. Passenger vehicles may fail to enter orbit (they require an additional small rocket burn half an orbit later, to circularize the orbit at destination altitude), and must be capable of reentry, but these will be not be fragile winged reentry. Instead, ablative heatshield slowdown reentries, followed and parachute splashdown in the ocean, perhaps 20 degrees west of the launch loop for vehicles intended for low earth orbit destinations, and 100 degrees west for GEObound vehicles.
Takeaway lesson: Reentry slowdown occurs 50 to 80 kilometers above the surface, and the slowdown forces double every 5 kilometers deeper into the atmosphere.
Mountains Don't Help
The uninformed have proposed launching rockets from tall mountaintops for decades; some propose launching from chemical cannons (or "modern" electrical cannons) attached to or drilled through them. I doubt that many of them have ever climbed a tall mountain, fired a cannon, built an electric motor, or launched a rocket into orbit. These proposals fail on physical and logistic grounds.
One of my favorite (and very strange) books is "Zero to Eighty" by "Akkad Pseudoman", a pseudonym for the Princeton Ph.D. physics professor, inventor, and entrepreneur Edwin Fitch Northrup. The book describes an electromagnetic accelerator (fast AC field and conductiveshelled vehicle) running up the side of Mt. Popocatépetl in southern Mexico, latitude 19°N, altitude 5.4 km.
Others have proposed Kilimanjaro (latitude 3°S, altitude 5.9 km), Chimborazo (latitude 1.5°S, altitude 6.3 km), or Everest (aka Sagarmāthā or Chomolungma, latitude 28°N, altitude 8.8 km) and other 7000m+ mountains in Asia.
I will leave out Mauna Kea (latitude 20°N, altitude 4.2 km) because that mountain is sacred to native Hawai'ians, and used by astronomers. I hope we can move the telescopes into space, and restore the native cemeteries on the summit.
7000 meter mountains seem impressive to 2 meter human beings, but they are trivial bumps compared to the depth of the atmosphere or the 12,740,000 meter diameter of the Earth. Scaled to the diameter of a 57mm billiard ball, these mountains would be 30 micrometers high; P500 finegrit polishinggrade sandpaper. Indeed, a 100 kilometer joyride to the "edge of space" is only half a millimeter compared to a billiard ball  beard stubble.
The excuse for spacelaunch schemes involving mountains is that the air is thinner at the top, somewhat less drag. But remember that the space shuttle Columbia was torn apart by aerodynamic forces at 70 kilometers altitude, where air density is 70 ppm of surface density. The air at the top of Mount Everest is 400,000 ppm of surface density, 6,000 times more air to punch through. Columbia was slowing from orbit, reentering, not attempting to punch out of the atmosphere with enough velocity remaining to go into orbit.
400 km altitude low earth orbits follow the curve of the earth, the equivalent of 2 millimeters above the surface a the billiard ball. Practically speaking, they are parallel to the surface. Mountains point diagonally upward; as we will see, the worst combination of two diametrically opposed problems.
On a round airless body like the Moon, an orbit can skim relatively close to the surface; the Apollo 11 command module orbited 62 kilometers above the Moon, and could have orbited down to 15 km ([citation needed], the perilune for the initial LM descent orbit) for an emergency rendezvous with a disabled lunar module. For comparison, the cruise altitude of the Concorde SST was 17 to 18 km, and the U2 flew at 21 km altitude.
Earth transfer orbits in vacuum, like the ascent of the lunar module, are zerogee ellipses; at the perigee and apogee portions of their orbits, they are also parallel to the surface of the body they orbit. They oscillate radially, smoothly, and their radial distance approximates a sine wave if the difference between perigee and apogee is relatively small.
If a mountaintop launch is into an elliptical transfer orbit (say, to LEO) that transfer orbit is parallel to the Earth's surface, and travels thousands of kilometers through very thick atmosphere. Much thicker atmosphere (6000 times denser) than spacecraft of average density use to slow down and reenter. Objects in thick atmosphere slow down extremely fast.
"Ah", I can hear you object, "of course we won't launch parallel to the surface, we will launch up the side of the mountain, going up through the atmosphere at a steep angle). Indeed, that reduces the time spent experiencing superhigh drag, so less velocity must be added up front to lose in the first seconds after release. But the path length is still shallow and long.
I don't know about the mountains of your imagination, but the tall mountains that I know about are rarely steeper than 40 degrees from base to peak. From the Khumbu ice fall to the peak of Everest is a 28 degree slope, 2.8 kilometers vertical rise in a 5.1 kilometer horizontal distance. A deep tunnel dug from Everest base camp in the west to the peak of Everest in the east would cover about 7.6 kilometers of horizontal distance, and 3.5 kilometers vertical distance, perhaps 8500 meters of curved then flat sloped acceleration track to stay on or in the mountain.

Space launch track in a tunnel bored through the west flank of Everest aka Chomolungma aka Sagarmāthā. Imagine the green line curving upward from horizontal, and the red line a straight diagonal tunnel slanting about 30 degrees upwards towards the summit. 
Good luck digging that tunnel, and connecting gigawatts of power lines to it.
A vehicle accelerating in a vacuum tunnel from 0 to 8000 meters per second over 8500 meters distance must accelerate at a = v^2 / 2 d = 8000² / 2 * 8500 = 3765 m/s², or 385 gees. You will spill your drink, as you are smashed into goo of the same consistency. If this is a 5000 kilogram vehicle, near the exit it will require a power level of P = m a v = 5000 * 3765 * 8000 = 150 gigawatts, 9% of all of China's 1650 gigawatts of electric generation for a few seconds.
I exaggerate  the power is pulsed, and will come from some gynormous energy storage system and be distributed through many gynormous switches and power cables. Which we must build and maintain deep in the frozen mountains at the end of yak trails. Right.
When the vehicle containing your liqufied remains leaves the vacuum tunnel, it will slam into a wall of 0.45 kg/m³ air at an altitude of 8.5 km (below cruise altitude for most jet passenger aircraft). Assuming a 1 square meter vehicle cross section times drag coefficient, the drag force is 0.45 * 8000² = 28.8 MegaNewtons, which will slow down the vehicle by 5760 m/s², or about 590 gees. Tough luck  in the remaining atmosphere, you will lose all your velocity and smack into the next mountain to the west of Everest.
Except you will vaporize first. Smashing into all that air at that speed compresses it, and it cannot move out of the way fast enough. That heats it  the drag heat is the drag force times the velocity, 230 gigawatts, radiated by a dense, thin layer of plasma rignt in front of the nose. Lucky for you, much of the energy is going to radiate to the front and sides, but you will still have on the order of 100 gigawatts per square meter heating your nose cone. For comparison, the surface of the Sun at 5778K emits 63 megawatts per square meter. Heat emission is proportional to the 4th power of temperature, so the nose cone will heat to 36000 degrees (Celsius or Kelvin, it really doesn't matter). The nose cone (and you) become a jet of incandescent plasma.
If you make the nosecone Really Pointy, the point will vaporize, then the blunter stub behind it, then ... you become a jet of incandescent plasma.
Wrong launch angle: Even if we send the atmosphere off to the cleaners to scrub out all US and Chinese electric generation exhaust, our troubles are not over. You are leaving the Earth at a 30 degree angle, not horizontal to the surface; you will be ber launched into an ellipse that intersects the Earth, about 180 degrees on the other side. your velocity is 8000 meters per second, but your horizontal velocity is "only" 8000 cos( 30⁰ ) = 6930 m/s. Using equations from this Server Sky page, we learn that the semimajor axis for a launch at circular orbit velocity but elevated above the surface by an angle of 30⁰ results in an eccentricity e = sin( 30⁰ ) = 0.5, to the perigee is (1e) a = 3193 km, and the apogee is (1+e) a = 9579 km. This orbit will arc above the earth for an apogee about 90⁰ to the east, then slam into the Earth's surface about 180⁰ to the east, on the far side of the earth.
With a large additional delta V at apogee, the perigee can be raised above the Earth, but sadly, your apogee insertion rocket motor will get smashed to goo by the launch acceleration. The most efficient apogee insertion motors are fragile, lowthrust, propellantthrifty electric thrusters, and will need weeks, not minutes, to raise orbit perigee.
Oh  and how are you going to do that accelerator? Railguns are ablative, and do more damage to the rails and power source than an ablative projectile will do to the target.
Switched coilguns ( later renamed "mass drivers" ) require vastly expensive high speed electronic switching to route impulse power from some storage source to create a fast moving magnetic field gradient ( very high dB/dt, very high voltage) that drives the projectile. The conductive projectile develops high currents to resist the field, which are proportional to the force; huge currents mean enormous ohmic heating (as it gets hotter, resistance increases, which makes it get hotter faster), which can melt or even vaporize the conducting portion of the vehicle.
To recap
 Mountains aren't nearly high enough to eliminate launch aerodynamic forces and reduce drag to humantolerable levels
 Mountains are way too short for zerotoorbital velocity change at humantolerable acceleration
 Single launch tracks up mountains cannot accomodate azimuth changes to different destination orbit inclinations
 note: neither can a single launch loop. High inclination orbits will require more launch loops.
 Angled launch tracks result in Earthintercepting orbits.
 Coilguns AKA mass drivers require way too much electrical power in a very short time, and the switches are way too expensive
 Railguns ditto, and the rails wear out too fast
People keep writing about this nonsense, as if aerodynamics and orbital geometry and electronic system design do not matter.
Alternate launch needs easily converted energy storage, a long horizontal launch path, and high vacuum between the exit and space.
That is why I looked for alternatives in the late 1970's and early 80's and after many failed attempts came up with the launch loop. Which may also be a failed attempt, but I spent 40 years, not 40 minutes, trying to find fatal showstoppers, instead finding many fascinating solutions for problems. Those solutions can be profitably applied to many other global problems.
I don't pretend to have all the answers for the Launch Loop, either, but there are no freshman physics showstoppers, and I am still finding improved answers. While many difficult advanceddegree problems remained to be solved, they are, as my late friend Robert Forward said, "mere engineering details". Increasing safety and efficiency and productivity, while reducing cost.
I have a few helpers with the details; currently, David Fierbaugh is helping with "acoustic climbers" to the stations, and the brilliant but overworked John Knapman has found some promising and robust control system methods. If you want to help solve the remaining fundamental problems (and get the solutions named after you), you'd better get involved soon, or the only thing your name will go on will be a ticket.