Question
How to calculate total change in pressure for a diver ascending from 20.6 meters depth to 4500 meters altitude
Answers
GPT-4o mini
To calculate the total change in pressure for a diver ascending from a depth of 20.6 meters underwater to an altitude of 4500 meters above sea level, you need to consider both the pressure changes due to water depth and atmospheric pressure changes with altitude.
### Step 1: Calculate the pressure at 20.6 meters underwater
The pressure underwater can be calculated using the formula:
\[
P = P_0 + \rho g h
\]
where:
- \( P \) = total pressure at depth (in pascals)
- \( P_0 \) = atmospheric pressure at sea level (approximately \( 101325 \, \text{Pa} \) or \( 1 \, \text{atm} \))
- \( \rho \) = density of water (approximately \( 1000 \, \text{kg/m}^3 \))
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = depth in meters (20.6 m)
Calculating the pressure underwater:
\[
P_{\text{water}} = 101325 \, \text{Pa} + (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(20.6 \, \text{m})
\]
Calculating the water pressure contribution:
\[
P_{\text{water}} = 101325 \, \text{Pa} + (1000)(9.81)(20.6)
\]
\[
P_{\text{water}} = 101325 \, \text{Pa} + 202,538.6 \, \text{Pa}
\]
\[
P_{\text{water}} = 303863.6 \, \text{Pa}
\]
### Step 2: Calculate the pressure at 4500 meters altitude
As the diver ascends to altitude, the atmospheric pressure decreases. The pressure at higher altitudes can be calculated using barometric formulas or by approximations, but a typical value for atmospheric pressure at an altitude of 4500 meters can be obtained from standard atmospheric tables, which is approximately 50,000 Pa to 60,000 Pa.
For simplicity, we can use \( P_{\text{atmospheric}} \approx 50000 \, \text{Pa} \).
### Step 3: Calculate the change in pressure
Now we can determine the total change in pressure from the underwater environment at 20.6 meters to 4500 meters altitude.
The diver starts at \( P_{\text{water}} = 303863.6 \, \text{Pa} \) and ends at approximately \( P_{\text{atmospheric}} = 50000 \, \text{Pa} \).
The total change in pressure is:
\[
\Delta P = P_{\text{final}} - P_{\text{initial}} = P_{\text{atmospheric}} - P_{\text{water}}
\]
\[
\Delta P = 50000 \, \text{Pa} - 303863.6 \, \text{Pa}
\]
\[
\Delta P = -253863.6 \, \text{Pa}
\]
### Conclusion
The total change in pressure for the diver ascending from 20.6 meters depth to 4500 meters altitude is approximately \( -253864 \, \text{Pa} \) (or about -253.86 kPa). This indicates a decrease in pressure as the diver ascends.
### Step 1: Calculate the pressure at 20.6 meters underwater
The pressure underwater can be calculated using the formula:
\[
P = P_0 + \rho g h
\]
where:
- \( P \) = total pressure at depth (in pascals)
- \( P_0 \) = atmospheric pressure at sea level (approximately \( 101325 \, \text{Pa} \) or \( 1 \, \text{atm} \))
- \( \rho \) = density of water (approximately \( 1000 \, \text{kg/m}^3 \))
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = depth in meters (20.6 m)
Calculating the pressure underwater:
\[
P_{\text{water}} = 101325 \, \text{Pa} + (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(20.6 \, \text{m})
\]
Calculating the water pressure contribution:
\[
P_{\text{water}} = 101325 \, \text{Pa} + (1000)(9.81)(20.6)
\]
\[
P_{\text{water}} = 101325 \, \text{Pa} + 202,538.6 \, \text{Pa}
\]
\[
P_{\text{water}} = 303863.6 \, \text{Pa}
\]
### Step 2: Calculate the pressure at 4500 meters altitude
As the diver ascends to altitude, the atmospheric pressure decreases. The pressure at higher altitudes can be calculated using barometric formulas or by approximations, but a typical value for atmospheric pressure at an altitude of 4500 meters can be obtained from standard atmospheric tables, which is approximately 50,000 Pa to 60,000 Pa.
For simplicity, we can use \( P_{\text{atmospheric}} \approx 50000 \, \text{Pa} \).
### Step 3: Calculate the change in pressure
Now we can determine the total change in pressure from the underwater environment at 20.6 meters to 4500 meters altitude.
The diver starts at \( P_{\text{water}} = 303863.6 \, \text{Pa} \) and ends at approximately \( P_{\text{atmospheric}} = 50000 \, \text{Pa} \).
The total change in pressure is:
\[
\Delta P = P_{\text{final}} - P_{\text{initial}} = P_{\text{atmospheric}} - P_{\text{water}}
\]
\[
\Delta P = 50000 \, \text{Pa} - 303863.6 \, \text{Pa}
\]
\[
\Delta P = -253863.6 \, \text{Pa}
\]
### Conclusion
The total change in pressure for the diver ascending from 20.6 meters depth to 4500 meters altitude is approximately \( -253864 \, \text{Pa} \) (or about -253.86 kPa). This indicates a decrease in pressure as the diver ascends.