A bridge is collapsing, and our poor MC is stuck in the middle. The ground beneath them collapses, but they miraculously are able to climb up the falling debris and safely make it to the main land before the whole bridge is lost.
My question is, how fast would one have to move to complete this feat, or one similar, with gravity equal to that on Earth's?
The important factor here is that human muscles can exert less force as they move faster. This paper shows that the elbow flexors are limited to speeds of around 6 m/s. This other study shows that the maximum velocity of most muscles is pretty similar, so we can assume that the maximum speed of the leg is similar. Since the femur is somewhat longer than the forearm, I'll assume a maximum speed of 7.5 m/s. Remember, at this speed the amount of force the legs can apply is zero.
Another important fact is that during a typical jump, the legs apply downward force for about 0.5 seconds.
Assuming that your character starts crouching on a large piece of rubble (so that the jumping force has negligible effect on the speed of the rubble), at the end of the jump the rubble will have a downward speed of about
$$ v = gt \approx 4.9~\text{m}/\text{s} $$
At this point, maximum force is reduced to 15–20%. A fit person can lift around 1.5× their weight in a squat, meaning that the 'isokinetic' (zero-speed) leg extension force is around 2.5× body weight. At the end of the jump the extension force will be reduced to around 0.4× body weight.
This means that even in the best-case scenario (starting squatted on a massive piece of rubble) a person cannot even stop themselves from falling, much less propel themselves upward.
In a more typical case, where your character starts standing, they will not be able to jump at all. When the ground under them starts to fall, both them and the ground accelerate downwards at the same speed. In a frame accelerating with the rubble, the character will appear weightless. When they flex their legs to jump, they will not move downward, but instead their feet will lift off of the ground. This is what happened when the Mythbusters tested this very scenario.
The Mythbusters tried this in one episode, and it was totally impossible for them to do (good thing they had a safety harness.)
The only realistic way for this to be possible is if the MC is some sort of superhero who can kick the falling debris downwards so fast he is essentially creating a rocket, with the debris as reaction mass. Since the debris is already falling downward at an acceleration of 1 "g" this will mess up the rocket equation (and someone smarter than me will have to do the math), but as a hand wave, if you want to "climb" the falling debris you would have to accelerate it by at least another 2 "g" (you would need 1 "g" just to remain in place and another "g" to accelerate upwards).
The downward spray of debris moving at 3 "g" will have a few consequences of its own, and if the debris runs out before the MC reaches the top, then he would somehow have to accelerate the air under his now rapidly spinning feet to make that last distance.
"Every action has an equal and opposite reaction". You would have to push down with enough force to push you yourself up. Does that sound easy? If it does, remember that that debris is accelerating away from you. Because of this, it is impossible.
Actually, the speed is not the problem. When your character pushes the falling debris to climb, the debris will fall faster because of the energy of his 'kick'. He won't go up even a bit because he and the debris are in freefall, and there's no normal force in freefall (normal force is basically the thing that pushes you up when you jump). So there's really not an answer to your question, because it is physically impossible, no matter the speed of the character.
On the other hand, if your character has some kind of jetpack, he can use it to 'fake' normal force and that way he could climb. But it wouldn't be a real climb, just a powered flight :P
More info about the world in which this is happening would be nice :D
Some sources: http://en.wikipedia.org/wiki/Normal_force
http://www.physicsclassroom.com/class/circles/Lesson-4/Weightlessness-in-Orbit
There is no fixed answer to how fast you would have to move: it depends on the speed at which the bridge is failing, both for individual elements tearing free and the transfer of failure to adjecent elements.
Imagine if you will: a deck unit groans loudly and then lurches as one steel cable breaks. This transfers force unevenly to the remaining cables so they failin turn. Cables stretch and frey; masonary crumbles around support joints; so the deck unit sinks, shifts, and lurches for a while. It doesn't instantly let go and start falling freely.
The characters don't jump from the free-falling block as some answers describe. The OP says they climb. They have some window of time to run and climb onto the adjecent deck unit. How much time? It depends on how fast the failure progresses. For greatest drama, it will fall as soon as the party makes it off, with one fellow left scrambling.
And it’s perfectly natural to suppose that moving the people's weight to the next deck unit will cause or potentate the failure of that one next.
This was a fun question to apply physics to! In short, your character has less than 0.9 seconds to jump up and grab the ledge. After that, they would have to hurry up and pull themselves up and run off the bridge to safety.
First, I found how high he/she would have to jump and grab onto a ledge with their fingers. A very good vertical jump is an astonishing 1.27 m. If they're 6 ft., then that gives them 6 ft, plus 2 ft of arms extended above them, plus their jump of 1.27 m. That gives a total distance of $d$ = 3.7 m. Plug into the following equation:
$d = v_{0} t + \frac{1}{2} a t^2$
Using $v_0$ = 0 m/s, $a$ = 9.8 m/s^2, and $d$ = 3.7 m, then $t$ = 0.87 s.
If your character is athletic, and quick to react, then I can see this as plausible. They can also more easily react if they hear the cracks forming so that that gives them an edge in reaction time.
Bonus, once he/she is up on the uncollapsed bridge, they would have to sprint the rest of the way. A Google search tells me the average sprinter (keyword athletic sprinter) can run 100 m in 14 seconds. So that's $v$ = 100 m / 14 s = 7.14 m/s. Find out the length of your bridge and use this formula: $d = vt$ to find out his/her distance or time to cross the bridge. Remember to divide the distance by 2 if the bridge collapses in the middle. Cheers!
No reasonable amount of speed will let you do that movie trick.
To jump up, you need to generate upward force with your jump. That requires that you push against something. But the debris is not solid ground, it will not provide you with any resistance to push up against.
The only thing you are pushing against, debris or not, is air and maybe, maybe you can generate a tiny amount of force through the inertia of the debris you are pushing against. However, without doing the math I'm quite sure any such gains are instantly eaten up by the air resistance you need to overcome on your way up.
