Problem 50 - University Physics with Modern Physics

Question

Puffy cumulus clouds, which are made of water droplets, occur at lower altitudes in the atmosphere. Wispy cirrus clouds, which are made of ice crystals, occur only at higher altitudes. Find the altitude \(y\) (measured from sea level) above which only cirrus clouds can occur. On a typical day and at altitudes less than 11 \(\mathrm{km}\), the temperature at an altitude \(y\) is given by \(T=T_{0}-\alpha y,\) where \(T_{0}=15.0^{\circ} \mathrm{C}\) and \(\alpha=6.0 \mathrm{C}^{\circ} / 1000 \mathrm{m} .\)

Step-by-Step Solution

Answer

Cirrus clouds occur above 2500 meters.

Step 1: Understand the problem

The exercise asks for the altitude above which only cirrus clouds (made of ice crystals) can form. This usually happens where temperatures are low enough for water droplets to freeze. We know the temperature decreases with altitude, given by the formula \( T = T_0 - \alpha y \). We need to find the minimum altitude \( y \) at which the temperature reaches 0°C, where freezing begins.

Step 2: Identify temperature formula components

The temperature formula is given as \( T = T_0 - \alpha y \). We know \( T_0 = 15.0^{\circ} \text{C} \) (temperature at sea level) and \( \alpha = 6.0^{\circ} \text{C}/1000 \text{m} \) (temperature gradient). We need to calculate when the temperature \( T \) becomes 0°C, which is a typical threshold for freezing and formation of cirrus clouds.

Step 3: Set up the equation to solve

To find the altitude \( y \) where the temperature drops to 0°C, set \( T = 0 \). Substitute \( T \), \( T_0 \), and \( \alpha \) into the formula:\[ 0 = 15.0 - 0.006 \times y \]

Step 4: Solve for altitude y

Rearrange the equation to solve for \( y \):\[ 0 = 15.0 - 0.006y \]\[ 0.006y = 15.0 \]\[ y = \frac{15.0}{0.006} \]Calculate \( y \):\[ y = 2500 \text{ m} \]

Step 5: Interpret the result

The calculated altitude of 2500 meters indicates where the temperature reaches 0°C, and thus the altitude above which only cirrus clouds can occur, since conditions allow water droplets to freeze into ice crystals beyond this point.

Key Concepts

Altitude and Temperature

As you rise in altitude, the temperature typically falls. This relationship is important in understanding weather patterns. The temperature at a specific altitude can be predicted using the formula:

• \( T = T_0 - \alpha y \)

Here, \( T_0 \) represents the temperature at sea level, while \( \alpha \) is the rate at which temperature decreases with altitude, known as the temperature gradient.
In the given exercise, \( T_0 \) is 15.0°C and \( \alpha \) equals 6.0°C per 1000 meters. This means that for every 1000 meters you ascend, the temperature drops by 6.0°C.
Understanding this relationship helps determine at which altitude specific weather phenomena, like the formation of different types of clouds, can occur.

Cloud Formation

Cloud types are classified based on their composition and the altitude at which they form.
Cumulus clouds, made up of water droplets, are generally found at lower altitudes where the temperature is warm enough to prevent freezing. In contrast, cirrus clouds, composed of ice crystals, form at high altitudes where temperatures are low enough to freeze water droplets.

• Puffy cumulus clouds occur in the lower altitudes.
• Wispy cirrus clouds develop at higher altitudes.

In this exercise, we established that at an altitude of 2500 meters, the temperature reaches 0°C, the freezing point. Thus, above this altitude, conditions are ripe for forming cirrus clouds as the colder temperatures cause water droplets to freeze into ice crystals.

Temperature Gradient

A temperature gradient refers to how quickly the temperature changes in a given distance. It is expressed here as \( \alpha = 6.0^{\circ} \text{C}/1000 \text{m} \), meaning the temperature falls by 6°C for every increase of 1000 meters in altitude.
Understanding temperature gradients is essential because they help predict how changing altitudes will affect weather conditions.

• Determines how drastically temperature drops with height.
• Affects cloud formation and other atmospheric phenomena.

In the problem, the temperature gradient played a crucial role in calculating the altitude where freezing occurs.
This principle also aids meteorologists in forecasting weather changes as the air moves to different atmospheric levels, altering temperature and subsequently, weather patterns.