Video Transcript
A submarine in the sea can dive to a depth where the pressure exerted by the water on the submarine is 10 atmospheres. The sea water has a density of 1025 kilograms per cubic meter. Find the maximum safe depth to which the submarine can dive, rounded to one decimal place, using a value of 101 kilopascals as the pressure of one atmosphere.
Here, we’re considering a submarine that’s diving deeper and deeper into the water until it gets to a point where the pressure on the submarine is equal to 10 atmospheres. For reference, one atmosphere is the pressure at sea level. The submarine is able to safely tolerate a pressure 10 times as great. We want to calculate the maximum safe depth, 𝑑, that the submarine can travel underwater. At this depth, we can imagine the whole column of water which is pressing down on the submarine.
In general, the pressure created by a fluid is equal to the density of that fluid times the acceleration due to gravity multiplied by the fluid height. In our case, that height is the depth 𝑑 we want to solve for. We can rearrange this equation so 𝑑 is the subject. It’s equal to the pressure exerted by the water divided by the water density multiplied by the acceleration due to gravity. Notice we’re ignoring the pressure due to the air column that’s on top of the water that’s on top of the submarine. This is because we’re told specifically that the maximum pressure exerted by the water on the submarine is 10 atmospheres. The pressure 𝑃 then is 10 atmospheres. Or written in units of kilopascals, it’s equal to 1010 kilopascals.
Moreover, we’re told that the density of sea water — we’ll call it 𝜌 — is 1025 kilograms per cubic meter. We can further recall that acceleration due to gravity, 𝑔, is 9.8 meters per second squared. Substituting all these values into our equation for the depth 𝑑, before we calculate this depth, there’s one more change to make so that all of our units are SI units on the same footing.
Let’s change the pressure from kilopascals into units of pascals. Since one kilopascal is 1000 pascals, if we add three zeroes to our pressure, then we’ll have the pressure in units of pascals. It’s 1010000 pascals. With our units set up like this, our final answer is in units of meters. When we round that result to one decimal place, we see that the maximum depth this submarine can safely reach is 100.5 meters.
