Calculating Mountain Height: A Physics Problem

Oct 21, 2025 by SLV Team 47 views

Hey guys! Let's dive into a classic physics problem: determining the height of a mountain. This kind of problem often pops up, and it's super important to understand the relationship between air pressure and altitude. We'll break down the question, go through the relevant concepts, and work towards a solution. The core idea is that as you go higher, the air pressure decreases. This decrease is due to the weight of the air above you. The higher you are, the less air is above you, and therefore, the lower the pressure. We'll use this principle, along with some physics formulas, to figure out how tall that mountain is. Get ready to flex those physics muscles! Understanding how air pressure changes with altitude is not only crucial for solving this particular problem but also has real-world applications. For instance, meteorologists use this principle to predict weather patterns, and pilots need to be aware of pressure changes when flying. So, let's get started. We'll cover the necessary formulas and provide a step-by-step approach to make sure everything is super clear.

The Problem: Unpacking the Details

Alright, let's look at the problem statement. The problem says: "If the air pressure at sea level ( $P_0$ ) is $101,300 ext{ Pa}$ and the air pressure on a mountain ( $P_h$ ) is $90,000 ext{ Pa}$ , determine the height of the mountain!" The problem gives us two crucial pieces of information: the air pressure at sea level and the air pressure at the mountain's peak. We're asked to find the height of the mountain. Before we jump into calculations, it's essential to understand the units involved. Air pressure is measured in Pascals (Pa), and height will be measured in meters (m). We need to remember that these units are compatible with the formulas we'll be using. Also, the problem implies that we're dealing with a simplified model. The real-world relationship between air pressure and altitude is affected by many factors, such as temperature, humidity, and even the type of air (i.e., whether it's dry air or moist air). However, for the sake of this problem, we'll assume a standard atmosphere to simplify things. So, what's our game plan? We will use a formula that connects air pressure and altitude. It's often based on the barometric formula, which you might have seen in a physics class. This formula relies on several constants and assumptions. In our simplified model, the temperature is considered constant, so the formula is a bit easier to apply. We'll need to know some constants, like the acceleration due to gravity and the density of air at sea level. Now, let's explore the key concepts and formulas.

Diving into the Physics: Concepts and Formulas

Here’s the deal: To solve this problem, we're going to use the barometric formula, which links air pressure to altitude. The simplified version of this formula is: $P_h = P_0 * e^{- (Mgh / RT)}$ , where:

$P_h$ is the air pressure at the mountain's height (in Pascals).
$P_0$ is the air pressure at sea level (in Pascals).
$e$ is the base of the natural logarithm (approximately 2.71828).
$M$ is the molar mass of air (approximately 0.02896 kg/mol).
$g$ is the acceleration due to gravity (approximately $9.81 ext{ m/s}^2$ ).
$h$ is the height of the mountain (what we want to find, in meters).
$R$ is the ideal gas constant (approximately $8.314 ext{ J/(mol*K)}$ ).
$T$ is the absolute temperature (in Kelvin). For this problem, we'll assume a standard temperature. The temperature varies with altitude, and that introduces additional complexity. We're going to keep it simple, which means we will assume a constant, standard temperature. We can also rearrange this formula to solve for height ( $h$ ), but it's important to keep track of each variable and make sure you understand what it represents. In the formula above, the exponential term describes how air pressure decreases with altitude. The negative sign indicates that pressure decreases as height increases. The term Mgh/RT is the ratio of potential energy to thermal energy. The ideal gas constant $R$ and the molar mass $M$ are fundamental constants in the study of gases. They help us relate the macroscopic properties of a gas (like pressure and temperature) to the microscopic properties of its constituent molecules. Remember, the barometric formula is based on several assumptions, such as a constant temperature and a uniform gravitational field. In reality, these parameters vary, but the formula provides a good approximation for calculating altitude. Let's get to the calculations now.

Calculations: Solving for Mountain Height

Okay, let's start crunching those numbers! We have the formula: $P_h = P_0 * e^{- (Mgh / RT)}$ . We need to isolate $h$ and solve for it. First, rearrange the formula to solve for $h$ . To do this, we can take the natural logarithm of both sides: $ln(P_h/P_0) = -(Mgh/RT)$ . Then, we can solve for $h$ : $h = - (RT/Mg) * ln(P_h/P_0)$ . We can plug in the values from the problem and the constants we know. The values are:

$P_0 = 101,300 ext{ Pa}$
$P_h = 90,000 ext{ Pa}$
$R = 8.314 ext{ J/(mol*K)}$
$M = 0.02896 ext{ kg/mol}$
$g = 9.81 ext{ m/s}^2$
$T$ (We will assume a standard temperature of 293 K, which is approximately 20°C). Remember, the temperature should be in Kelvin. Let's put all of the variables in the formula and do the math: $h = - (8.314 ext{ J/(mol*K)} * 293 ext{ K} / (0.02896 ext{ kg/mol} * 9.81 ext{ m/s}^2)) * ln(90,000 ext{ Pa} / 101,300 ext{ Pa})$ . Now, you just need to calculate the value of $h$ ! The result is approximately 1,027 meters. Remember to pay close attention to the units. They need to be consistent to make the calculation right. In this case, we end up with meters for the height. Double-check your calculations, especially the value of the natural logarithm. It’s super important to make sure everything lines up. Let's move on to the interpretation of our result.

Interpreting the Result and Conclusion

Alright, guys, we got a height of approximately 1,027 meters for the mountain. This means that, based on the air pressure difference and our simplified model, the mountain is about 1,027 meters high. The answer provides a valuable understanding of the relationship between air pressure and altitude, and it can be used to solve other related physics problems. Remember that the accuracy of this result depends on the assumptions we made, such as a constant temperature. In the real world, atmospheric conditions are much more complex. The temperature can vary significantly with altitude, affecting air pressure. We also assumed a standard atmosphere, which is a simplified model. For more accurate calculations, you might need to use more advanced models and consider factors like temperature gradients. However, for a basic physics problem, this method provides a solid and useful approximation. So, we've successfully calculated the height of the mountain! We've also learned about the barometric formula, the relationship between air pressure and altitude, and the importance of simplifying assumptions in physics. Well done, everyone! Now, you're all set to tackle similar problems. Keep practicing! Remember to review all steps, from understanding the problem to making assumptions, selecting the correct formula, calculating, and interpreting the results. These skills are essential not only for physics but for solving problems in many different fields. Great job, and keep up the amazing work!