Question

Many altimeters determine altitude changes by measuring changes in the air pressure. An altimeter that is designed to be able to detect altitude changes of \(100 \mathrm{~m}\) near sea level should be able to detect pressure changes of a) approximately I Pa. d) approximately \(1 \mathrm{kPa}\). b) approximately 10 Pa. e) approximately \(10 \mathrm{kPa}\). c) approximately \(100 \mathrm{~Pa}\).

Short answer

Answer: e) approximately 10 kPa

Step-by-step solution

Understand the Barometric Formula

The Barometric Formula gives the relationship between pressure and altitude in Earth's atmosphere: \(\Delta P = -P_0 \cdot \rho \cdot g \cdot \Delta h\) Where \(\Delta P\) is the pressure change, \(P_0\) is the reference pressure (pressure at sea level), \(\rho\) is the density of air, \(g\) is the acceleration due to gravity and \(\Delta h\) is the altitude change.

Values at sea level

At sea level, the pressure (\(P_0\)) is approximately \(101325 \textrm{Pa}\), the density of air (\(\rho\)) is approximately \(1.225 \textrm{kg/m}^3\), and the acceleration due to gravity (\(g\)) is approximately \(9.81 \textrm{m/s}^2\).

Calculate the pressure change

Plug the given altitude change (\(\Delta h = 100 \textrm{m}\)) and the sea level values into the Barometric Formula and solve for the pressure change, \(\Delta P\): \(\Delta P = -(101325 \textrm{Pa}) \cdot (1.225 \textrm{kg/m}^3) \cdot (9.81\textrm{m/s}^2) \cdot (100 \textrm{m})\) \(\Delta P = -12150 \textrm{Pa}\) Since pressure decreases with an increase in altitude, the negative sign indicates a decrease in pressure. However, we are only interested in the magnitude of the pressure change, so we ignore the sign: \(\Delta P = 12150 \textrm{Pa}\).

Compare the results with the options given

Since 12150 Pa is roughly equal to \(12.15 \thinspace kPa\), the altimeter should be able to detect pressure changes around \(10 \mathrm{kPa}\). So, the correct answer is: e) approximately \(10 \mathrm{kPa}\).

Key concepts

Barometric Formula

The Barometric Formula is an essential tool for understanding how atmospheric pressure varies with altitude. It encapsulates the principle that as one moves up in altitude, the weight of the air above decreases, hence the pressure drops. It's expressed mathematically as the change in pressure \( \Delta P \) being the product of the initial pressure \( P_0 \) at a reference point, usually sea level, the density of the air \( \rho \) at that level, the acceleration due to gravity \( g \) and the altitude change \( \Delta h \) as the following equation shows:
\[ \Delta P = -P_0 \cdot \rho \cdot g \cdot \Delta h \]
This formula helps us predict how pressure will decrease as we gain altitude. In contexts like aviation, it's vital for calibrating instruments such as altimeters which rely on the relationship between air pressure and altitude to provide accurate readings. A solid understanding of the Barometric Formula is a stepping stone for those interested in fields related to meteorology, aviation, and even mountain climbing.

Pressure Altitude Changes

Altimeters are designed to measure altitude by interpreting pressure changes in the atmosphere. When we look at pressure altitude changes, it's important to grasp how sensitive these devices are to fluctuations in air pressure. For example, an altimeter that can identify a 100-meter change near sea level would have to detect a pressure change in the range of 10 kilopascals (kPa).
The larger the altitude change \( \Delta h \), the more significant the pressure change \( \Delta P \). However, the rate of change in pressure is not linear but decreases with altitude due to the thinning atmosphere. Therefore, altimeters need to be calibrated with the nonlinear nature of pressure changes with respect to different altitudes to maintain their accuracy throughout various atmospheric conditions.

Density of Air

The density of air, represented as \( \rho \), is a critical factor in understanding atmospheric phenomena and is especially relevant when applying the Barometric Formula. At sea level, air is denser due to the cumulative weight of the air above it. However, as one ascends, the density decreases – this is because the air particles are spread further apart at higher altitudes.
Variations in air density can be attributed to changes in altitude, temperature, and moisture content. For instance, denser cold air contains more molecules per unit volume than warmer air. This factor contributes significantly to pressure changes – detecting a 100-meter altitude change near sea level involves measuring a pressure change that takes into account both the initial pressure and density of air at the starting level. Thus, in practical applications like calibrating an altimeter, it's essential to factor in the standard air density at sea level, which is roughly \( 1.225 \mathrm{kg/m}^3 \). This value helps in translating pressure changes to altitude readings accurately.