Elad’s blog

Over six months ago, I presented you this small riddle, about the elephant on the lake. I was surprised to find that it still bothers some of my friends; last night, it seems, Lior wasted precious sleeping hours trying to proove a mathematical  solution. So for all who are still interested, here is my solution and proof; do try to solve it yourself before looking at the answer. I included a bit of simple math in there, and I apologize for being so unpolite; but it is the easiest way to convince some people that this is really the solution.

The main principle that matters for this case is the famous Archimedes’ law. It is a basic principle of hydrostatics, and states the concept of buoyancy: a body submerged in fluid feels a buoyant force acting upon it, that equals to the weight of the displaced fluid. Some of you may know the famous story about Archimedes of Syracuse and how he discovered it while taking a bath, to the astonishment of the Syracusian girls.
Now let’s analyze the situation. Assume that the raphsody has a density of _r and a volume of V_r, which gives it a total mass of M_r=_r x V_r. The elephant also has a density and a volume, so let’s denote M_e=_e x V_e. The water of the lake, of course, have their own density _w.

In the initial condition, when the elephant stands on the raphsody. There is an equilibrium, so the gravity that tries to push both the elephant and the raphsody down is perfectly balanced by the buoyancy caused by the displacement of the water. The water level at the lake is a bit higher then it would have been if there was no raphsody in it. What is the volume of water displaced in the initial situation? exactly the volume necessary to balance the gravity:

_w x V_initial=_r x V_r+_e x V_e

V_initial=(_r x V_r+_e x V_e)/_w

Now let’s analyze the second situation. Once the elephant is inside the water, the amount of water it displaces depends on its density relative to the water: if it is less dense then the water it will float, only partially submerged, but if it is much denser then water it will sink all the way to the bottom and even press on it, because the buoyant force is not enough to balance its own weight. The raphsody still displaces the same amount of water due to its internal weight. So the total volume of water displaced in the final position equals:

V_final=(_r x V_r+min(_{e, w)} x V_e)/_w

The minimum function in the last equation takes care of the case the elephant is floating. Now let’s separate the three possible cases:

1. If the water is denser then the elephant, the elephant is partially floating, and the volume of water displaced in the final state equals to that in the initial state, hence the water level doesn’t change.
2. If the density of the water and the elephant is exactly the same, also the water level doesn’t change. By the way, this is roughly the real case.
3. If the density of the elephant is bigger then the density of the water, it will sink down to the bottom, but displace water only relatively to its volume, which is less then the amount of water it displaced in the initial state. As less water gets displaced, the water level of the lake will decrease.

So, the final answer is that: if the elephant jumps from the raphsody to the water, the water level in the lake will either stay the same or decrease, depending on the density of the elephant; it would never increase.

Just a few side notes:

• The raphsody doesn’t matter here. We intrinsicly assumed that we can neglect the amount of air displaced by the elephant at the initial stage, this air also presses on the water on the final stage, but as the elephant is much denser it doesn’t matter. As a nice exercise, ask yourself what would happen if this wasn’t the case.
• We assumed a constant density for all factors. This, of course, is not the case. Due to hysrostatic pressure the water at the bottom of the lake are denser then those near its surface. There are some nice experiments to show it, or you can just dive and feel it yourself. Anyhow, the result wouldn’t change - only the math would get a bit more complicated.
• Arik, in his comment, was almost right: the splash would help to decrease the water level in the lake, but even without this effect the water level may decrease.
• Lior, in the detailed answer he sent me privately, was also almost right: congratulations!

And one last remark… Try to compare your initial intuition with what the physics say. Ain’t it nice?

This entry was posted on Thursday, March 22nd, 2007 at 9:13 and is filed under Life. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.