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## Jumping in a falling lift

If the cable holding up a lift snaps and you plummet earthward, can you survive by jumping just before you hit the ground? This sounds like a job for Newtonian physics!

Let’s set up the problem. Let:

$h_{fall}$ = the height from which the elevator falls.

$h_{surv}$ = the maximum height from which you could fall onto solid ground and survive more often than not

$h_{jump}$ = your vertical leap height i.e. how high can you jump?

Now, in the absence of wind resistance (more on that later), we can relate these quantities to relevant velocities (a.k.a. speed; g = 9.8m/s^2 = acceleration due to gravity):

$v_{fall} = \sqrt{2gh_{fall}}$ = the velocity of the elevator just before it hits the ground.

$v_{surv} = \sqrt{2gh_{surv}}$ = the maximum impact velocity that you could survive

$v_{jump} = \sqrt{2gh_{jump}}$ = your velocity as you leave the ground, jumping vertically.

Jumping (assuming that your mass is much less than that of the lift) means that the velocity at which you hit the ground is reduced from $v_{fall}$ to $v_{fall} - v_{jump}$. You survive if this number is less than $v_{surv}$. Putting it all together, jumping will allow you to survive a fall from a height of

$h_{fall,max} = (\sqrt{h_{surv}} + \sqrt{h_{jump}})^2$

For example, if you could ordinarily survive a fall of 25 metres, and can perform a vertical leap of 0.5 metres, then you could survive an elevator fall of 32.5 metres.

However, the assumption of zero wind resistance for the falling lift probably isn’t reasonable. The lift is likely to reach terminal velocity, slowed by the compression of the air in the elevator shaft. In this case, we would need the terminal velocity to be less than

$v_{terminal} = (\sqrt{2gh_{surv}} + \sqrt{2gh_{jump}})$

In our example, a 25 metre (ballistic) fall is a velocity of 80km/h. A perfectly timed jump would allow you to survive a terminal velocity of 91km/h.

In conclusion, modern elevators have a wide variety of safety backups – it isn’t just a single cable. But if the worst should happen, you might as well …

### One Response

1. OK, a few things here.

Firstly jumping in free fall can actually be quite hard.

Secondly, assuming you can jump while the elevator is falling, I think we need to look at the downsides here as well. Can one really time one’s “jump” accuartely enough to be sure it will have a positive effect on your survival? Because if you get it wrong then you would increase the harmful effects of the fall. One jumps by accelerating onesself upwards – the same thing that kills you when the lift stops abruptly, and then you do. So, jumping as the lift hits the ground will add to the acceleration you experience and splatter you more effectively.

Thirdly, Jumping (or is it really better to call it “pushing off” – due to the free fall aspect?) basicaly just gives you a small constant velocity relative to the falling lift (think astronauts in the space station). Any advantge you might get from your “jump” occurs between the moment you leave the lift floor and the moment you arrive at the top of the lift and then decelerate. Of course the faster you push off the shorter this time will be, so clearly a compromise here based on how accurately you are able to co-ordinate your push with the lift impact.

Clearly an area that needs further research 🙂