How Does Mathematics Explain the Mechanism of a Cartesian Diver?

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The Cartesian diver operates based on the principles of pressure and volume as described by the equation P1*V1=P2*V2, reflecting the behavior of gases in a closed system. When pressure is applied to the bottle, it compresses the air inside the diver, causing the volume of air to decrease and the density of the diver to increase. As the density surpasses that of water, the diver sinks. The discussion also raises questions about the relative air pressure inside the container compared to the external environment before pressure is applied. Understanding these mathematical relationships is key to explaining the mechanism of a Cartesian diver.
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i would like to know how a cartesian diver works using mathematics. I have found how the diver works using words and explanation but not using equations and math. Please HELP!
 
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P1*V1=P2*V2 Since you basically have a closed system (the diver plus the water bottle) by applying pressure on the bottle, you are effectively transferring this pressure to the cartesian diver. This is because water is pretty much uncompressible. So, because the pressure is increased, the volume of air must decrease. Because we have a closed system, the water goes into the void created by the compressed air, the density increases until it is greater than that of water and it sinks.
 
Hey I am new here and i also need help on the Cartesian Diver. I was just wondering if The air pressure is the same, less than, much less than, greater or much greater than the air pressure outside of the bottle or container before pressure is applied to the container.
 

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