Parachuting inside bales of hay

[geoffrey159]

Homework Statement

During World War II the Russians, lacking sufficient
parachutes for airborne operations, occasionally dropped soldiers
inside bales of hay onto snow.
The human body can survive an average pressure on impact of
30 lb/in2 . Suppose that the lead plane drops a dummy bale equal
in weight to a loaded one from an altitude of 150 ft, and that the
pilot observes that it sinks about 2 ft into the snow. If the weight of
an average soldier is 144 lb and his effective area is 5 ft2 , is it safe
to drop the men?

Homework Equations

energy conservation

The Attempt at a Solution

For clarity, I set h1 = 150ft, h2 = 2ft, mg = 144lb, and S = 5lb/ft^2.

Using conservation of energy, the paratrooper's speed just before impact is $v_i = \sqrt{2gh_1}$.

If $E_r$ is the mechanical energy once the paratrooper is at rest 2ft under snow, $E_i$ the mechanical energy at impact, and $\vec N$ the force of snow on man, then:

$\left\{ \begin{array}{} E_r = 0\\ E_i = mg (h_1+h_2) \\ E_r - E_i = \int_{\vec r_{impact}}^{\vec r_{rest}} \vec N.d\vec r \end{array} \right.$

And, because the force of the snow is vertical,

$mg(h_1+h_2) = \int_{0}^{h_2} N(y) dy \approx h_2 N_{av}$

So the average pressure the paratrooper must resist is

$\frac{N_{av}}{S} = 144*152/(2*5) \ lb/ft^2 = 152/(2*5) \ lb/in^2 = 15.2 \ lb/in^2 < 30 \ lb/in^2$

So yes it is 'safe' (non lethal). Is it correct ?

 

[mfb]

Under the (problematic) assumption that the force is constant over the whole length, and if the airplane magically stops in the air, yes.
I'm sure the Russians used a better unit system.

 

[geoffrey159]

I agree it is not very realistic :-)
Thank you for the reply !

 

[haruspex]

Under the (problematic) assumption that the force is constant over the whole length, and if the airplane magically stops in the air, yes.
I'm sure the Russians used a better unit system.

Yes, two good points. The horizontal speed might not be a big deal if the bales are round, though of course that leads to the separate question of how great a rotational acceleration is tolerable.
For the constancy of vertical force, I suspect that depends largely on snow depth. For very deep snow it could be fairly constant.

 

[HalfLostAndSearching]

I would say that your solution is logically sound and uses appropriate equations and assumptions. However, it is difficult to determine the accuracy without more specific information, such as the density and compressibility of the snow, the exact weight and distribution of the soldier's equipment, and the exact size and shape of the bales of hay. It may also be important to consider the potential for injury from impact forces, even if the pressure is within a safe range. Overall, while this solution may provide a reasonable estimate, it would likely need to be tested and refined in a controlled experiment before it could be considered definitively correct.