Spider-Man, of mass 90kg, is perching 10 meters above the ground when he notices his (depending on the continuity) crush/girlfriend/wife Mary Jane (50 kg) being menaced by… I dunno, a menace. I’m more of a DC fan really.
He swings down toward her to spirit her our of harm’s way. Assuming he starts from rest and lets gravity do the work, how fast is he going just before he reaches her? Well, he has a gravitational potential energy mgh. And just before he reaches her, it’s all been turned into kinetic energy, half the mass times velocity squared.
Which works out to about 14 m/s. Now right after he reaches her, clearly they’ll be going slower. It’s a collision though, so we can’t assume all the kinetic energy stays conserved. But momentum will be. That means if we want to know how fast they’re going as a unit after he catches her, the answer will be
v2 being the velocity after the rescue, the m with the mj is Mary Jane’s mass. So I figure that gives us a post collision velocity of 9 m/s.
And now the kinetic energy associated with that speed is going to turn back into potential energy as they swing upward in their arc. How high will it take them? Clearly not as far, they lost energy in the collision and even if they hadn’t they’re still having to split the energy between them.
So for the v we just found and their new maximum height h, we find
Which gives the height they’ll reach on the upswing as about 4.1 meters. I’m not sure that would be adequate to get them out of harm’s way. Let’s hope it’s one of the low-rent villains. Otherwise it’s probably a job for Superman!
(Yes, this was a brief concept quiz I gave my students.)