Problem 25 - Basic Engineering Mathematics

Question

The pressure \(p\) of the atmosphere at height \(h\) above ground level is given by \(p=p_{0} \mathrm{e}^{-h / c}\), where \(p_{0}\) is the pressure at ground level and \(c\) is a constant. Determine the rate of change of pressure with height when \(p_{0}=10^{5}\) Pascals and \(c=6.2 \times 10^{4}\) at 1550 metres.

Step-by-Step Solution

Answer

The rate of change of pressure with height at 1550 meters is approximately -1.57 Pascals per meter.

Step 1: Understand the Formula

We have the formula for atmospheric pressure at height given by \(p = p_0 e^{-h / c}\). Here, \(p_0\) is the pressure at ground level and \(c\) is a constant related to the atmosphere's properties.

Step 2: Differentiate the Formula

To find the rate of change of pressure with respect to height, we need to differentiate \(p = p_0 e^{-h/c}\) with respect to \(h\). The derivative of \(e^{-h/c}\) with respect to \(h\) is \(-\frac{1}{c} e^{-h/c}\), so:\[\frac{dp}{dh} = p_0 \left( -\frac{1}{c} e^{-h/c} \right) = -\frac{p_0}{c} e^{-h/c}.\]

Step 3: Substitute Known Values

We substitute the known values: \(p_0 = 10^5\) Pascals, \(c = 6.2 \times 10^4\) m, and \(h = 1550\) m into the derivative to find the rate of change:\[\frac{dp}{dh} = -\frac{10^5}{6.2 \times 10^4} e^{-1550 / 6.2 \times 10^4}.\]

Step 4: Simplify the Exponent

Calculate the value of the exponent:\[-\frac{1550}{6.2 \times 10^4} \approx -0.025.\]

Step 5: Calculate the Exponential Term

Compute the value of the exponential term:\[e^{-0.025} \approx 0.9753.\]

Step 6: Final Rate of Change Calculation

Substitute the value of the exponential term:\[\frac{dp}{dh} = -\frac{10^5}{6.2 \times 10^4} \times 0.9753 \approx -1.57 \, \text{Pascals per meter}.\]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that describes the rate at which a function changes at any given point. It's like understanding how fast you're driving at a specific moment. This rate of change is what we call the derivative. In mathematical terms, it's the slope of the tangent line to the function at a given point.
When you differentiate a function, you are essentially finding a new function which gives you the slope at any point on the original curve. For linear functions, this is straightforward as the slope is constant. However, for more complex functions, including exponential functions, differentiation is crucial to understand dynamic changes.

• The factor "-1/c" arises from the chain rule applied to exponential functions.
• The final derivative, \(-\frac{p_0}{c} e^{-h/c}\), tells us how the pressure decreases with height.

Knowing how to calculate a derivative allows us to solve real-world problems involving rates of change, like how atmospheric pressure changes with altitude.

Exponential Functions

Exponential functions are a type of mathematical function where the variable is an exponent. These functions are widely used in real-world applications because they model growth or decay processes very effectively. Examples include population growth, radioactive decay, and finance.
In the context of atmospheric pressure, exponential functions describe how pressure decreases as you rise higher in altitude. The function \(p = p_0 e^{-h/c}\) is a classic example where pressure decreases exponentially with height.

• The base of the exponential function in this case is e, a mathematical constant approximately equal to 2.718.
• e is the natural base because it naturally arises in contexts involving continuous growth or decay.

This function showcases how even slight changes in height can significantly impact atmospheric pressure, illustrating the power and importance of exponential functions in modeling natural processes.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air in the Earth's atmosphere. At sea level, it's about 101325 Pascals or 1013.25 hPa. As you ascend, the air becomes less dense, and thus the pressure decreases.
This pressure drop with increasing altitude is predictable with an exponential function, where the rate of decrease is influenced by properties like temperature and the composition of the atmosphere.

• At 1550 meters, pressure changes can be precisely calculated using the differentiated function.
• Understanding this concept is crucial for fields such as meteorology, aviation, and even personal activities like hiking.

Being able to calculate the rate of pressure change with height allows scientists and engineers to make informed predictions and decisions based on atmospheric conditions.