Problem 9 - College Physics

Question

In the troposphere (the lowest part of the atmosphere), the temperature decreases rather uniformly with altitude at a so-called "lapse" rate of about \(6.5^{\circ} \mathrm{C} / \mathrm{km}\). What are the temperatures (a) near the top of the troposphere (which has an average thickness of \(11 \mathrm{~km}\) ) and (b) outside a commercial aircraft flying at a cruising altitude of \(34000 \mathrm{ft} ?\) (Assume that the ground temperature is normal room temperature.)

Step-by-Step Solution

Answer

(a) -51.5°C; (b) -47.34°C at cruising altitude (34,000 ft).

Step 1: Introduction to the Problem

We are given a ground temperature (normal room temperature), an altitude, and a lapse rate of 6.5°C/km. Our task is to calculate the temperature at different heights in the troposphere due to this lapse rate.

Step 1: Convert Ground Temperature to Celsius

Room temperature is often assumed to be approximately 20°C. We need to use this as our starting point to apply the lapse rate.

Step 2: Calculate Temperature at the Top of the Troposphere

The top of the troposphere is at 11 km. Using the lapse rate of 6.5°C/km, we find the temperature change by multiplying the altitude by the lapse rate.\[\text{Temperature change} = 11 \times 6.5 = 71.5^{\circ} \text{C}\]We subtract this temperature change from the ground temperature:\[\text{Temperature at the top} = 20^{\circ} \text{C} - 71.5^{\circ} \text{C} = -51.5^{\circ} \text{C}\]

Step 3: Convert Aircraft Cruising Altitude to Kilometers

The cruising altitude of the aircraft is 34,000 feet. We need to convert this to kilometers. 1 foot is approximately 0.0003048 kilometers.\[34,000 \text{ feet} \times 0.0003048 \approx 10.36 \text{ km}\]

Step 4: Calculate Temperature at Aircraft Cruising Altitude

Now that the aircraft's altitude is about 10.36 km, we calculate the temperature change using the lapse rate.\[\text{Temperature change} = 10.36 \times 6.5 = 67.34^{\circ} \text{C}\]Subtract this from the ground temperature:\[\text{Temperature at 34,000 ft} = 20^{\circ} \text{C} - 67.34^{\circ} \text{C} = -47.34^{\circ} \text{C}\]

Step 6: Conclusion

The calculations determine the temperature at different altitudes in the troposphere by applying the lapse rate to the room temperature. We found both the temperature at the top of the troposphere and at the altitude of a typical commercial aircraft.

Key Concepts

Lapse Rate

The lapse rate is a key concept to understand how temperature changes with altitude, especially in the troposphere, which is the lowest layer of Earth's atmosphere. It represents the rate at which the temperature decreases as we ascend.

The standard lapse rate is approximately 6.5°C per kilometer. This means that for every kilometer you go up, the temperature drops by 6.5°C. This uniform decrease is crucial for calculating temperatures at various altitudes.

• Example: At 2 kilometers above the ground, the temperature is expected to be 13°C cooler than the ground temperature.

Understanding the lapse rate allows meteorologists to predict weather patterns and helps aviators to anticipate temperature changes when flying at different altitudes.

Altitude Conversion

Altitude conversion is essential when calculating temperature changes at different altitudes because we often use different measurement units such as feet or meters. When dealing with aviation or geographical data, you might receive altitudes in feet, while scientific calculations might require kilometers.

To convert from feet to kilometers, you multiply the number of feet by approximately 0.0003048. This conversion factor comes from the relationship that 1 foot equals 0.3048 meters.

• Example: 34,000 feet is approximately 10.36 kilometers.

This conversion ensures that you are using consistent units of measure, which is critical for accurate temperature calculations using the lapse rate.

Temperature Calculation

Calculating temperature at different altitudes involves using the lapse rate effectively. Start with a known ground temperature, typically assumed to be around 20°C if we consider room temperature.

Then, determine the altitude difference from the ground to the point of interest. Multiply this altitude by the lapse rate to find the temperature change. Finally, subtract this change from the ground temperature to find the new temperature.

• Formula: Temperature at altitude = Ground temperature - (Altitude in km × Lapse rate)
• Example: For an altitude of 11 km, the temperature would decrease by 71.5°C, making the temperature at that height -51.5°C.

By following these steps, you can determine the expected temperatures at various heights within the troposphere.

Atmosphere Layers

The Earth's atmosphere is composed of several layers, each having distinct characteristics and temperature patterns. These layers include the troposphere, stratosphere, mesosphere, thermosphere, and exosphere.

The troposphere is the lowest layer, extending up to about 11 km above the Earth's surface. It's where most of our weather occurs and contains the majority of the atmosphere's mass. This layer exhibits the lapse rate we discussed, with temperatures decreasing as altitude increases.

• Troposphere: Up to 11 km, weather, aircraft flying zone.
• Stratosphere: 11 km to 50 km, home of the ozone layer.

Each layer beyond the troposphere exhibits different behaviors with respect to temperature changes, and understanding these layers can provide insights into atmospheric phenomena such as weather systems and climate change.