Seeing a lot of good posts now
I agree with everything you are saying
Michael, it all makes sense. When you're saying that the answer to why Vy is the best option is obvious I completely agree. It will leave you both more potential and kinetic energy, so basically you have more altitude and speed to play with.
However, let's say you choose to turn back above a determined "safe turn back altitude" of 600 feet. At 600 feet in a C172 standard day with lets say a 5000 foot runway, you will make it back no problem. However, lets say you climb at Vy until you reach 1200 feet (straight out departure). Will you then make it back?
Lets say it took you 2 minutes to get to 1200 feet. Since the climb angle is shallower than the glide angle, after a certain amount of time on departure (assuming no winds), you probably won't reach the runway. What if you climbed at Vx for 2 minutes? Would you then make it? You would be lower (maybe 1000 feet), but you would perhaps be close enough to be able to make it back (since 1000 feet is way more than enough to safely make the turn back). Therefore I wonder if above a certain altitude, Vx might be better if you choose to turn back to the runway (again, all depending on atmospheric conditions, which is what makes this so complicated since they are always different). Quick example: two days ago with a student taking off from a 5000 foot runway we were maybe 3000-4000 feet from the runway end when we reached 600 feet. Yesterday with another student taking off from a 6000 foot runway we were right above the runway end when we reached 500 feet. Big difference.
Of course, the problem is that the engine never quits at a certain altitude or above a certain altitude. If I knew that the engine always quit above safe altitude turn back altitude, it might make sense to fly at Vx since we will be closer to the runway. But if the engine quits below safe altitude or in any other case where flying straight ahead is the only option, Vy is obviously the best (for the same reasons you specify: more energy).
PA, I like your calculations! And you are right, at 500 feet per minute there's no way you will make it back to the runway. However, in an airplane that climbs better (C172 will easily do more than 500 ft/min standard atmospheric conditions) or more favorable atmospheric conditions it's a different story.
And
Michael, I think what
hawk37 meant to say was that V2 approximates Vxse and not Vyse. I guess Venr would be more approximate to Vyse.