Answers to the suitcase problem

If you are on a boat and toss a suitcase overboard, will the water level rise or fall?

Normally, we wouldn't be interested in the rise or fall of the water level in a body of water large enough to permit boating. In fact, tossing a suitcase out of a boat is not going to change the water level perceptibly. The question is whether it changes the level in principle.

We can try and imagine the problem. Anytime you throw weight out of a floating boat, the boat gets lighter and rises. However, that's not what this question asks. It asks whether the water level rises or falls.

If the suitcase were to be magically removed or evaporated into thin air, the answer becomes more straightforward because the boat would simply be lighter and float higher. This makes the water level decrease.

However in this question, the suitcase is tossed out of the boat, and into the water. This begs us to question further. What if the suitcase sinks? What if the suitcase floats? Will the answer be the same for both cases?

In the case of the suitcase sinking, pretend you throw a heavy suitcase overboard, having first secured it to the boat with fishing line. The boat briefly rises, then the line draws taut as the suitcase sinks as far as the line permits. This makes the boat dragged down by the weight of the suitcase. The displacement of the boat plus suitcase is identical to what it was originally. If you snip the fishing line, the suitcase will sink to the bottom while the boat's hull rises upward. This decreases the volume displaced and hence lowers the water level.

In the case of the suitcase floating, we would be more interested in the weight displaced. In school, we learnt about Archimedes principle, which states that the upthrust acting on an object is equal to the weight of the fluid displaced. If I have a 5kg beach ball floating in a bathtub, the beach ball displaces 5kg of water. Similarly, if I have five 1kg beach balls floating in the same body of water, the total weight of water displaced is still 5kg. This means that the volume displaced is also the same. Hence, if the suitcase floats the water level will remain the same.

Conclusion: tossing a suitcase off a boat makes no difference in water level if the suitcase floats. If the suitcase sinks, the water level falls.