Question

The barometer of a mountain hiker reads 750 mbars at the beginning of a hiking trip and 650 mbars at the end. Neglecting the effect of altitude on local gravitational acceleration, determine the vertical distance climbed. Assume an average air density of \(1.20 \mathrm{kg} / \mathrm{m}^{3}\).

Short answer

Answer: The mountain hiker climbed approximately 849.34 meters during the hiking trip.

Step-by-step solution

Write down the hydrostatic pressure equation

We can find the height difference using the hydrostatic pressure equation for fluids: \(\Delta{P} = -\rho g \Delta{h}\) where \(\Delta{P}\) is the change in pressure during the hike (in pascals), \(\rho\) is the air density (in kg/m^3), \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{m} / \mathrm{s}^2\)), and \(\Delta{h}\) is the height difference (in meters) that we need to find.

Convert the pressure readings to pascals

The initial and final pressure readings are given in millibars (mbars). We need to convert them to pascals, as they are the SI unit for pressure: 1 mbar = 100 Pa Initial pressure (P1) = 750 mbar * 100 Pa/mbar = 75000 Pa Final pressure (P2) = 650 mbar * 100 Pa/mbar = 65000 Pa

Calculate the change in pressure

Now we need to calculate the change in pressure (\(\Delta{P}\)) during the hike: \(\Delta{P}\) = P1 - P2 = 75000 Pa - 65000 Pa = 10000 Pa

Use the hydrostatic pressure equation to find the height difference

We can now rearrange the hydrostatic pressure equation and plug in the values of \(\Delta{P}\), \(\rho\), and \(g\) to find the height difference \(\Delta{h}\): \(\Delta{h} = -\frac{\Delta{P}}{\rho g} = -\frac{10000 \mathrm{Pa}}{1.20 \mathrm{kg} / \mathrm{m}^{3} * 9.81 \mathrm{m} / \mathrm{s}^2} = \frac{10000 \mathrm{Pa}}{11.772 \mathrm{kg} / \mathrm{m}^{3} \cdot \mathrm{m} / \mathrm{s}^2} \approx 849.34 \mathrm{m}\) The mountain hiker climbed approximately 849.34 meters during the hiking trip.

Key concepts

Barometric Pressure

Understanding barometric pressure is crucial when attempting to determine changes in altitude, such as during a mountain hiking trip. Barometric pressure, also known as atmospheric pressure, is the force exerted onto a surface by the weight of the atmosphere above that surface. It is an important factor in weather systems and is also used in various scientific and engineering calculations.

At sea level, the standard barometric pressure is 1013.25 millibars or hectopascals. As altitude increases, barometric pressure decreases since there is less air above to exert pressure. This concept is precisely what allows hikers with barometers to estimate their elevation gain or loss during a trek. Pressure is measured in various units, and for scientific purposes, the Pascal (Pa) is the standard unit - where 1 mbar equals 100 Pa. Knowing how to convert from one unit to another, as seen in the exercise with the hiker's pressure readings, is fundamental for accurate calculations.

Air Density

Air density plays a vital role in a wide range of phenomena, from weather patterns to the calculations required for aviation and assessing altitude changes during a hike. Air density, denoted by the symbol \( \rho \), is the mass per unit volume of Earth's atmosphere. It is often measured in kilograms per cubic meter \(\mathrm{kg} / \mathrm{m}^3\).

Air density is affected by temperature, pressure, and humidity. Warmer air tends to be less dense than cooler air, and moist air is less dense than dry air. For the exercise involving the hiker, a constant average air density of \(1.20 \mathrm{kg} / \mathrm{m}^{3}\) is assumed, which simplifies the calculation of the vertical distance. However, in reality, air density can change significantly over different altitudes and temperatures, which can impact the accuracy of such calculations.

Vertical Distance Calculation

Calculating vertical distance, or elevation gain, is of particular interest to hikers, engineers, and geographers among others. Knowledge of the change in vertical distance is essential in fields ranging from aviation to meteorology.

In our scenario, the hydrostatic pressure equation, \( \Delta{P} = -\rho g \Delta{h} \), where \(g\) is the acceleration due to gravity, is used to find the vertical distance climbed by the hiker. Solving for \( \Delta{h} \) in the equation allows us to compute how high the hiker has climbed based on the change in barometric pressure. The inclusion of air density and gravity in the calculation allows for a more accurate result, assuming that other environmental variables are kept constant.

For those who struggle with understanding vertical distance calculations, visualizing the problem can often help. Consider imagining the height as the space between two clouds or the difference in altitude between two flat plains atop mountains with changing pressures. Remember, as you ascend or descend, pressure changes, and using the hydrostatic pressure equation, we can quantify that change in terms of vertical distance, the result of which gives us a measure of our elevation change during the hike.