Air Temperature As dry air moves upward, it expands and, in...

Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about $1^{\circ} \mathrm{C}$ for each $100-\mathrm{m}$ rise, up to about $12 \mathrm{km}$. (a) If the ground temperature is $20^{\circ} \mathrm{C}$, write a formula for the temperature at height $h$ (b) What range of temperatures can be expected if a plane takes off and reaches a maximum height of $5 \mathrm{km} ?$

Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about $1^{\circ} \mathrm{C}$ for each $100-\mathrm{m}$ rise, up to about $12 \mathrm{km}$.
(a) If the ground temperature is $20^{\circ} \mathrm{C}$, write a formula for the temperature at height $h$
(b) What range of temperatures can be expected if a plane takes off and reaches a maximum height of $5 \mathrm{km} ?$

Key Concepts

Lapse Rate

The lapse rate is a measure of the rate at which air temperature decreases with an increase in altitude. It is a key concept in atmospheric science and meteorology, particularly when examining how temperature variations affect weather and climate. A typical value for the dry adiabatic lapse rate is about 1°C per 100 meters, which provides a simplified linear estimation of temperature change in the lower atmosphere.

Linear Relationship

In many real-world applications, temperature decreases with height can be modeled using a linear function. A linear relationship involves a constant rate of change, represented by the slope, and a starting value, represented by the intercept. This makes it straightforward to express temperature as a function of altitude, where the ground temperature is the intercept and the lapse rate serves as the slope.

Function Notation and Evaluation

Function notation is used to succinctly express the relationship between variables, such as temperature and height in this context. By defining a temperature function T(h) that gives the temperature at any given altitude h, one can easily calculate and analyze the effects of altitude on temperature. Evaluating this function at specific heights, such as at the ground level or the maximum altitude of an aircraft, further allows for the determination of temperature variations over that interval.

Transcript

00:01 Okay, so the first thing i want to do here, or the first information that we're giving is that the temperature decreases by 1 degrees celsius for each 100 meters that we climb in altitude.

00:14 Now, we need all of our units to be the same.

00:19 So for temperature, we have 1 degrees celsius, and then we're giving 20 degrees celsius, so we're good.

00:24 Now, we're given 100 meters for the increase in temperature, but when everything else talks about height, we're given kilometers.

00:31 So we need to find a way to either convert the kilometers to meters or the meters to kilometers.

00:37 I'm going to convert the meters to kilometers.

00:41 So to do that, for 100 meters, it's equal to 0 .1 kilometers.

00:48 So we're saying that the temperature increases 1 degrees celsius per 10th of a kilometer.

00:56 So furthermore, if i want to express this in terms of 1 kilometer, if i multiply both of these by 10, 10, i get for every one kilometer, the temperature increases or decreases by 10 degrees celsius.

01:11 So for every kilometer that we climb in the atmosphere, the temperature will get 10 degrees colder.

01:19 So that can help us with our formula.

01:22 We have t is equal to, we start off with our original temperature, which is 20.

01:27 We're subtracting, but we're getting lower, so we have minus.

01:30 And then it's going down about 10 degrees per kilometer.

01:35 And the kilometers are height and kilometers...

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