Super capacitors break the sound barrier.

[Geek-9pm]

Did that headline get your attention?
That analogy was made by a researcher earlier this year after a break through  was made in super capacitor technology.
Without going into the deep, deep technical explanation, super capacitors now are the best deal available for start storing large amounts of electrical energy.
Some reports this year indicate that researchers may accomplish their goals and be able to deliver super capacitors at a reasonable cost to replace almost any other kind of electrical storage system, including batteries that are commonly used in cars and trucks. It is said super capacitors have much more power per pound than anything else. That means they could be used to power small aircraft and devices like drones where power and weight are a large consideration.

Looking into the future we might live in cities where electrical power is the only kind of power used in the city. Combustion would never be used to power anything. All types of mobile devices would have super capacitors that could run vehicles for hours at a time before needing a recharge at an electrical station.
There are a number of different links available to support this kind of reasoning. Rather than try to post them here, you can try looking them up yourself. Just do a search on the expression "super capacitors" and see what you find. You should find some articles, including at least one from the IEEE, about the new advance in super capacitor technology.

So far, there's been lots of speculation. But large super capacitors at economical prices are not yet a reality. But if you want, smaller super capacitors are available for experimental use or some practical applications. Be aware that even when you have a low voltage super capacitor, it is capable of creating an arc that can l do a lot of damage to your face if you don't have protective glasses. Apparently that is why some experimenters put a small device in series with the super capacitor they used for experiments. The small device is some kinds of resister with a very low value that limits the discharge current and reduces the amount of spark you get if you accidentally short leads together.
 
EDIT: You do not need to buy anything or join any group.  Here is a free line to  the IEEE paper  about Super Capacitors:
http://spectrum.ieee.org/green-tech/fuel-cells/nitrogen-supercharges-supercapacitors
(Skip the ads.)

[Salmon Trout]

So far, supercapacitors are quoted as storing between 0.01 and 0.04 MJ/Kg (million Joules per kilogramme). The paper you linked to says their devices "could" store 41 watt-hours per kilogram. That's 0.17 MJ/Kg. The same as lead-acid batteries. So maybe engine starting and ignition duty in automobiles if they can make them cheap and durable enough.

It is said super capacitors have much more power per pound than anything else.

Anything else? Other ways of storing energy (MJ/Kg):
Hydrogen (compressed at 700 bar) 142
Gasoline 46
Li-on battery 0.4–0.9

[Geek-9pm]

Thank you Salmon Trout. The comparison was about electrical energy.
Sometime Energy and Power are thought of as the same thing. In formal physics, they are not.

In physics, power is the rate of doing work. It is the amount of energy consumed per unit time. Having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second (J/s), known as the watt in honour of James Watt, the eighteenth-century developer of the steam engine.

https://en.wikipedia.org/wiki/Power_(physics)
So then, super  capacitors have the advantage of quick start. But not the endurance.

By the way, you are right on about compressed air. It has a big promise.
http://gazettereview.com/2016/09/zero-pollution-motors-update-happened-shark-tank/
The above indicates it might be just hot air.   

[Salmon Trout]

By the way, you are right on about compressed air. It has a big promise.

I didn't mention compressed air, it was hydrogen I quoted, and the energy isn't from the compression. Compressed air energy storage has about the same energy desnity per unit weight as lead-acid batteries.