Cron and chaps:
There several ways to get your head round why the ski-jump improves a Harrier STO performance without getting into the maths. But some may not be familiar with the basics of the VTO and STO so forgive me for tidying those away first.
A Harrier will only VTO if its weight is less than some 96 percent of the thrust available. (Not 100% as you need a tad to spare to accelerate the thing upwards as well as carrying the weight)
So, at max AUW you will typically be short of three or four tons of lift (actual amount depends on the mark of motor and airframe) This calls for an STO where you need to go fast enough on the surface for the wing to carry the excess weight plus some margin to stop it staggering, then when you bang the nozzles down the thing will get airborne on a mix of engine and wing lift.
Now imagine doing this from a flat deck. Life is better than from a runway as the deck height above the water is free and so you can delete the margin needed to stop the runway stagger. You shoot off into the air regardless and can also rotate the aircraft to the optimum AOA – which you cannot do ON the runway because of the bicycle main gear being well aft of the CG.
But all of this ship advantage vanishes (and then some) in rough water. If the ship is pitching you may get to the end of the deck when the bow is down and finish up diving towards the sea. Pulling out from this dive requires the ability to pull g, which means a considerable margin of lift is needed. So, once you are using all the deck to accelerate to as high an 'end' speed as possible you have no option but to reduce STO weight. And this weight reduction can be very large so quickly negating the benefit of the STO in the first place. So you become a fair weather only toy.
Now stick a ramp on the end where the exit angle is greater than any conceivable bow down tendency and you make every launch start with a climb courtesy of the ramp and the gear. End of ship motion problem Now consider a ramp angle much greater than that needed for the above and you really start to win. Your question is Why?
The secret is the ramp gives you TIME in the air even if you have zero lift.(I trust you can accept that even your bike or car would fly for quite a few moments before it gradually fell back towards the sea if you shot up a ramp at seventy or so kts)
If your bike or car is now a Harrier - where the engine can carry say 70 percent of the weight - then even without ANY wing lift you will only sink back towards the sea at 30% of gravity. If you arrange that the wing lift actually carries half of that 30% lift deficit then gravity only sucks you down at 15% of its normal rate. Which means you curve upwards for many seconds before you get level, let alone sink. You use this TIME to accelerate the wing to a higher airspeed (through the proper use of nozzle angle) to get more aerodynamic lift and so fly happily away. Even though you started at a weight that could NEVER have flown off a flat deck in a calm sea – because then you would not have had enough acceleration time before you got to the water.
There are many other advantages to the ski-jump in terms of handling and safety but your question was about performance.
Hope that helps.
John