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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 …