If you're talking physics in a debate between hard floor and spring floor, a spring floor is safer in every aspect of the landing.
Let me explain...
Say you have a watermelon. You go to the local pool and you go all the way up to the high dive. You stand at the very edge of the high dive, hold out the watermelon and let it drop into the pool. What happens? Nothing. The watermelon makes a huge splash but other than that, the watermelon comes out unscathed. Now, take that same watermelon, go back up to the high dive and hold the watermelon out again, but this time, let it drop onto the cement right below. What happens? The watermelon shatters on impact. Why? Because the cement does not "give" into the watermelon upon landing. The water, on the other hand does, and that "give" is what saves the watermelon. Lets say the watermelon falls at 9 miles per hour. So when falling on concrete, the acceleration of our watermelon goes from 9mph to 0mph in 0 seconds. When it hits the water, it goes from 9mph to 0mph in something like 0.4 seconds. Although the 0.4 seconds does not seem like a lot, its enough to save the watermelon.
Now what do watermelons have to do with this debate? Switch the cement and water with the hard and spring floors. Granted that both will absorb the impact of the athlete whether they are jumping, tumbling, dancing or falling, the spring floor, with its mat, ply-wood and springs, will extend the time that the acceleration is stopped compared to that of a hard floor, thus saving the athlete from harm.
If you are talking about other aspects of cheer, like the added height of jumping or tumbling or the rebound you get from the spring floor, that's going to be situational. At least the falling issue is going to be somewhat the same all the way around. Whether you fall on your head, knees, back, or arm from a stunt, jump or tumbling pass, the outcome on a spring floor will always be better than the outcome from the hard floor.
Is there another performance surface that is safer than a spring floor? Not to my knowledge because anything softer would inhibit the athletes performance. We just participate in a dangerous sport....
@
kingston - is that what you were looking for?