Why not build an Electric Plane?

Lots of people are already trying this. It is indeed "technologically feasible" as you state, but does not yet appear to be commercially feasible. The weight of the batteries required to produce reasonable speed and range results in very little payload available. But it'll surely be done, eventually as technology continues to improve.

I'd prefer that Tesla and SpaceX both stay out of that market, however. There are no real synergies with either of their current business models or markets, and they are both already attacking huge challenges without yet any assurance of long-term success. Let them focus on becoming completely sustainable and successful before they start diversifying into further challenges.

There is a solar powered plane that was sponsored by a group of Swiss businesses.
It came thru Dallas in May I believe, was out at DFW airport for a week or so, had wing span of a 747, single passenger, powered by four 10 HP electric motors.
This one is going across the country.
They are deep into development for a slightly (15%) large version that is supposed to go around the world.
It is nearly a glider, because it really rises and floats on the upper atmosphere winds.

Limits of the plane are more what a MAN can do, because a similar plane could be flown more effectively as a drone, not worried about being tired, food, water, waste, etc.

Related thread:

http://www.teslamotorsclub.com/showthread.php/1307-Electric-planes

agelston said:

Seriously. Musk states in his hyperloop design "A quiet supersonic plane immediately solves every long distance city pair..."

a 50/50 SpaceX + Tesla joint venture should start developing small, private electric planes.

Click to expand...

Emphasis added.

The main problem here is creating a supersonic propeller plane. It hasn't been done, and has serious problems associated with reaching Mach 1. You almost certainly would have to have two propellers (one sub sonic, and one supersonic) to even begin to contemplate this type of approach.

Basically as your plane starts approaching super sonic speeds the plane starts tearing the propeller apart with shockwaves.

While I hardly ever say "never", this is a rare case where I'd say we'll "never" see a supersonic propeller-driven plane. Even with prop planes going 200-250 mph, we have problems with the prop tips approaching Mach 1, and the shock waves both destroy efficiency and increase sound. Prop and supersonic don't go together and I don't think they ever will. Quiet didn't even enter the equation.

Propellers are more efficient than jets (basic physics says it's more efficient to move high volume at low speed than low volume at high speed), and that's why most people fly props... they're much cheaper to operate. But they're slower. Prop planes face serious challenges when attempting to go very, very fast: the fastest propeller-driven planes do go 550 mph in air racing, but that's racing and not commercial transportation. They use ungodly amounts of power, like 3000 HP for a single-seat airplane, while I used to fly a twin-engine plane that used 400 HP for six people in comfortable seats. Plus, 550 mph is still only like Mach 0.80 or something at Reno's altitude.

Supersonic airplanes are, as far as I can see, always going to be jets until we have some new and totally-different form of propulsion.

Elon Musk is already in discussion with Stark Industries about it:
Elon Musk in Iron Man 2 - YouTube