So, I went to the bank yesterday and as it's common these days, the waiting line was huge. While waiting for my number to be called, I overheard two kids talking about someone jumping from a falling rock right before it hit the ground. Their mother (I assume) soon lost patience regarding the ridiculous amount of people waiting and left, but the image of The Coyote pulling that stunt remained in my head so I took out a piece of paper.
Now, I'm aware cartoons are not the best place to find logical answers to physic related questions but... What about a situation like that in real life? Say a man is walking carelessly on the street and suddenly a hole opens up and a huge block of pavement just falls in it, taking him with it. If he could determine the exact moment before both him and the pavement would hit the bottom, would jumping right then help in any way?
Yes, but probably in a negligible way.
The easiest approach to this is paying attention to the velocities.
Let's say our guy falls from a height of 50m.
stop laughing at my drawing please
Both he and the rock are in free fall (accelerated only by gravity), so the (modulus of the) velocity just before it reaches the bottom, if he doesn't jump, is the same as the pavement's and is given by:
Now, they say a human jumping gains a speed of about 3 m/s and I believe that's plausible so, if our guy jumps right before the chunk of pavement hits the bottom, he'd have a velocity of -3 m/s relative to the falling piece of pavement. If we assume there's an inertial frame of reference in the bottom of the pit, his final velocity would be:
Where Va is the absolute velocity of our man right before hitting the ground, Vrel is the relative velocity of the man in reference to the pavement and Vp the final velocity of the pavement. The absolute velocity then would be:
Well, he'll hit the ground at 28 m/s, that's approximately 100 kph (about 62 mph)! So, in our scenario of a 50 meter fall, he doesn't make it.
Now, what if he was falling from 15 meters instead? In that case his final absolute velocity would be:
That's equivalent to 50 kph and 30 mph. Not a pretty hit, but he has much better probabilities of surviving this one!
Depending on the height of the fall, jumping might help with mitigating the impact once in the bottom.
Please note that a bunch of assumptions were made here. Technically, while on free fall over the pavement, our guy would find it really difficult to jump, since trying to bend his knees would result in his legs going up while his torso goes down a similar distance due to conservation of momentum. Also, on the actual fall, inertia would retard his fall from the pavement an instant. And we didn't even mention other factors like wind resistance. Either way, I think this is a funny approach to the subject.
Also, if someone finds there's a mistake in the reasoning or in the math (as simple as it was) please let me know!
