Free Fall – Would It Help to Jump Inside a Free-Falling Elevator?

accelerationcollisionequivalence-principlefree fallnewtonian-gravity

Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When?

Best Answer

While everyone agrees that jumping in a falling elevator doesn't help much, I think it is very instructive to do the calculation.

General Remarks

The general nature of the problem is the following: while jumping, the human injects muscle energy into the system. Of course, the human doesn't want to gain even more energy himself, instead he hopes to transfer most of it onto the elevator. Thanks to momentum conservation, his own velocity will be reduced.

I should clarify what is meant by momentum conservation. Denoting the momenta of the human and the elevator with $p_1=m_1 v_1$ and $p_2=m_2 v_2$ respectively, the equations of motion are

$$ \dot p_1 = -m_1 g + f_{12} $$ $$ \dot p_2 = -m_2 g + f_{21} $$

Here, $f_{21}$ is the force that the human exerts on the elevator. By Newton's third law, we have $f_{21} = -f_{12}$, so the total momentum $p=p_1+p_2$ obeys

$$ \frac{d}{dt} (p_1 + p_2) = -(m_1+m_2) g $$

Clearly, this is not a conserved quantity, but the point is that it only depends on the external gravity field, not on the interaction between human and elevator.

Change of Momentum

As a first approximation, we treat the jump as instantaneous. In other words, from one moment to the other, the momenta change by

$$ p_1 \to p_1 + \Delta p_1, \qquad p_2 \to p_2 + \Delta p_2 .$$

Thanks to momentum "conservation", we can write

$$ \Delta p := -\Delta p_1 = \Delta p_2 .$$

(Note that trying to find a force $f_{12}$ that models this instantaneous change will probably give you a headache.)

How much energy did this change of momentum inject into the system?

$$ \Delta E = \frac{(p_1-\Delta p)^2}{2m_1} + \frac{(p_2+\Delta p)^2}{2m_2} - \frac{p_1^2}{2m_1} - \frac{p_2^2}{2m_2} .$$ $$ = \Delta p(\frac{p_2}{m_2} - \frac{p_1}{m_1}) + (\Delta p)^2(\frac1{2m_1}+\frac1{2m_2}) .$$

Now we make use of the fact that before jumping, the velocity of the elevator and the human are equal, $p_1/m_1 = p_2/m_2$. Hence, only the quadratice term remains and we have

$$ (\Delta p)^2 = \frac2{\frac1{m_1}+\frac1{m_2}} \Delta E .$$

Note that the mass of the elevator is important, but since elevators are usually very heavy, $m_1 \ll m_2$, we can approximate this with

$$ (\Delta p)^2 = 2m_1 \Delta E .$$

Energy reduction

How much did we manage to reduce the kinetic energy of the human? After the jump, his/her kinetic energy is

$$ E' = \frac{(p_1-\Delta p)^2}{2m_1} = \frac{p_1^2}{2m_1} - 2\frac{\Delta p\cdot p_1}{2m_1} + \frac{(\Delta p)^2}{2m_1}.$$

Writing $E$ for the previous kinetic energy, we have

$$ E' = E - 2\sqrt{E \Delta E} + \Delta E = (\sqrt E - \sqrt{\Delta E})^2 $$

or

$$ \frac{E'}{E} = (1 - \sqrt{\Delta E / E})^2 .$$

It is very useful to estimate the energy $\Delta E$ generated by the human in terms of the maximum height that he can jump. For a human, that's roughly $h_1 = 1m$. Denoting the total height of the fall with $h$, we obtain

$$ \frac{E'}{E} = (1 - \sqrt{h_1/h})^2 .$$

Thus, if a human is athletic enough to jump $1m$ in normal circumstances, then he might hope to reduce the impact energy of a fall from $16m$ to a fraction of

$$ \frac{E'}{E} = (1 - \sqrt{1/16})^2 \approx 56 \% .$$

Not bad.

Then again, jumping while being weightless in a falling elevator is likely very difficult...