Problem 8 - Fundamentals of Physics

Question

The bends during flight. Anyone who scuba dives is advised not to fly within the next \(24 \mathrm{~h}\) because the air mixture for diving can introduce nitrogen to the bloodstream. Without allowing the nitrogen to come out of solution slowly, any sudden air-pressure reduction (such as during airplane ascent) can result in the nitrogen forming bubbles in the blood, creating the bends, which can be painful and even fatal. Military special operation forces are especially at risk. What is the change in pressure on such a special-op soldier who must scuba dive at a depth of \(20 \mathrm{~m}\) in seawater one day and parachute at an altitude of \(7.6 \mathrm{~km}\) the next day? Assume that the average air density within the altitude range is \(0.87 \mathrm{~kg} / \mathrm{m}^{3}\).

Step-by-Step Solution

Answer

The pressure change is approximately 245847 Pa.

Step 1: Calculate Pressure at Diving Depth

To find the pressure at the depth of 20 m in seawater, we use the equation for hydrostatic pressure: \[ P = P_0 + \rho g h \]where \( P_0 \) is the atmospheric pressure at sea level (\( 101325 \mathrm{~Pa} \)), \( \rho \) is the density of seawater (approximately \( 1025 \mathrm{~kg/m^3} \)), \( g \) is the acceleration due to gravity (\( 9.81 \mathrm{~m/s^2} \)), and \( h \) is the depth (20 m). Substituting these values, we get:\[ P = 101325 + (1025 \times 9.81 \times 20) \approx 101325 + 200790 = 302115 \mathrm{~Pa}\]

Step 2: Calculate Pressure at Altitude

Next, calculate the pressure at 7.6 km altitude using the barometric formula approximation:\[ P = P_0 \exp\left(\frac{-\rho gh}{P_0}\right) \]where \( \rho \) is the average air density (\( 0.87 \mathrm{~kg/m^3} \)), and \( h \) is the height above sea level (7.6 km). Substituting the values:\[ P = 101325 \exp\left(\frac{-0.87 \times 9.81 \times 7600}{101325}\right) \approx 101325 \exp(-0.644) \approx 56268 \mathrm{~Pa} \]

Step 3: Calculate Change in Pressure

Now, determine the change in pressure experienced by the soldier moving from 20 m underwater to 7.6 km in the air:\[ \Delta P = P_{\text{dive}} - P_{\text{altitude}} = 302115 - 56268 \approx 245847 \mathrm{~Pa} \]

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at a given depth due to the weight of the fluid above. It plays a crucial role in understanding forces underwater:
When you dive into the ocean, you experience hydrostatic pressure.
This pressure depends on the depth of the water and its density.
The deeper you go, the more pressure you feel. The formula to calculate hydrostatic pressure is given by:

• \( P = P_0 + \rho g h \), where:
  • \( P_0 \) is the atmospheric pressure at sea level.
  • \( \rho \) is the fluid's density (seawater in our example).
  • \( g \) is the acceleration due to gravity.
  • \( h \) is the depth below the surface.

This knowledge is essential for scuba divers to ensure safety and avoid the bends.

Barometric Formula

The barometric formula is used to model how atmospheric pressure decreases with altitude. Understanding this concept is pivotal for calculating pressure at different heights above sea level. The formula is an exponential function:

• \( P = P_0 \exp\left(\frac{-\rho gh}{P_0}\right) \), where:
  • \( P_0 \) represents the initial atmospheric pressure at sea level.
  • \( \rho \) is the air density.
  • \( g \) is the acceleration due to gravity.
  • \( h \) denotes the height above sea level.

This calculation is significant when considering pressure changes in weather systems or during high-altitude flights, like that of a parachuting soldier.

Scuba Diving Physics

Scuba diving involves intricate physics, revolving around pressure changes and gas laws. Underwater, divers breathe air at increased pressures, which affects gases in their bloodstream:

• As a diver descends, the water pressure increases, causing the air mixture in their tanks to compress.
• Nitrogen, part of the air mixture, dissolves into the blood, increasing as pressure rises.
• Divers must ascend slowly to prevent rapid decompression and the formation of nitrogen bubbles, which can cause the bends.

Safety guidelines, such as not flying after diving, help prevent decompression sickness and ensure diver safety while transitioning to lower pressure environments.

Altitude Pressure Effects

Understanding altitude effects on pressure helps in many practical scenarios, from aviation to weather forecasting. As altitude increases, atmospheric pressure decreases:

• At higher altitudes, the air is thinner, resulting in lower air pressure.
• This can affect breathing, as less oxygen is available in each breath.
• For divers, flying soon after diving increases risk because lower pressure can cause dissolved gases in the blood to form bubbles.

Knowing these effects is critical for activities at different altitudes, ensuring safety and physiological well-being during transitions between altitudes, like a skydiving mission.