Problem 3 - University Physics with Modern Physics

Question

(a) On January \(22,1943,\) the temperature in Spearfish, South Dakota, rose from \(-4.0^{\circ} \mathrm{F}\) to \(45.0^{\circ} \mathrm{F}\) in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was \(44.0^{\circ} \mathrm{F}\) on January \(23,1916 .\) The next day the temperature plummeted to \(-56^{\circ} \mathrm{F} .\) What was the temperature change in Celsius degrees?

Step-by-Step Solution

Answer

Spearfish: 27.2°C increase; Browning: 55.6°C decrease.

Step 1: Convert Initial Fahrenheit Temperature to Celsius (Spearfish)

Use the formula \[ C = \frac{5}{9}(F-32) \] to convert the initial temperature of \(-4.0^{\circ} F\) to Celsius. Substitute: \[ C = \frac{5}{9}(-4.0 - 32) = \frac{5}{9}(-36) = -20^{\circ} C \]

Step 2: Convert Final Fahrenheit Temperature to Celsius (Spearfish)

Convert the final temperature of \(45.0^{\circ} F\) to Celsius using the same formula. Substitute: \[ C = \frac{5}{9}(45.0 - 32) = \frac{5}{9} \times 13 = 7.2^{\circ} C \]

Step 3: Calculate Temperature Change in Celsius (Spearfish)

Subtract the initial Celsius temperature from the final Celsius temperature: \[ 7.2^{\circ} C - (-20^{\circ} C) = 7.2^{\circ} C + 20^{\circ} C = 27.2^{\circ} C \]

Step 4: Convert Initial Fahrenheit Temperature to Celsius (Browning)

Convert the initial temperature of \(44.0^{\circ} F\) to Celsius. Substitute: \[ C = \frac{5}{9}(44.0 - 32) = \frac{5}{9} \times 12 = 6.7^{\circ} C \]

Step 5: Convert Final Fahrenheit Temperature to Celsius (Browning)

Convert the final temperature of \(-56^{\circ} F\) to Celsius. Substitute: \[ C = \frac{5}{9}(-56 - 32) = \frac{5}{9} \times (-88) = -48.9^{\circ} C \]

Step 6: Calculate Temperature Change in Celsius (Browning)

Subtract the final Celsius temperature from the initial Celsius temperature: \[ 6.7^{\circ} C - (-48.9^{\circ} C) = 6.7^{\circ} C + 48.9^{\circ} C = 55.6^{\circ} C \]

Key Concepts

Fahrenheit to Celsius Conversion

To convert temperatures from Fahrenheit to Celsius, you use a straightforward formula:

• Start with the Fahrenheit temperature.
• Subtract 32 from it.
• Multiply the result by \( \frac{5}{9} \).

This formula arises because of the different scales that Fahrenheit and Celsius use for freezing and boiling points of water. While water freezes at \(0^{\circ}C\), it does so at \(32^{\circ}F\).
In the example of Spearfish, to convert \(-4^{\circ}F\) to Celsius, you do \(-4 - 32 = -36\), then multiply by \(\frac{5}{9}\), which results in \(-20^{\circ}C\).
A similar process converts \(45^{\circ}F\) to \(7.2^{\circ}C\). This simple conversion is essential when comparing temperatures on different scales.

Temperature Change Calculation

Calculating the change in temperature is another simple yet crucial operation. To find out how much the temperature has increased or decreased:

• Convert both the initial and final temperatures to Celsius, if they are not already.
• Subtract the initial temperature from the final temperature.

If the temperature rises, the result is positive, indicating a gain. If it falls, the result is negative, indicating a loss.
In Spearfish, the temperature rose from \(-20^{\circ}C\) to \(7.2^{\circ}C\), resulting in a change of \(7.2 - (-20) = 27.2^{\circ}C\), indicating a significant increase. In Browning, a drop from \(6.7^{\circ}C\) to \(-48.9^{\circ}C\) means a fall of \(55.6^{\circ}C\), a major decrease that underscores the extreme weather of the area.

Historical Weather Events

Historical weather events often highlight dramatic temperature changes, which can be astonishing and offer insights into local climate conditions.
For example, Spearfish, South Dakota, holds the record for one of the fastest recorded temperature changes. On January 22, 1943, it went from \(-4^{\circ}F\) to \(45^{\circ}F\) in just two minutes, likely due to a sudden shift in air masses. Similarly, in Browning, Montana, there's another historical instance where temperatures plummeted sharply within a day, highlighting the unpredictable nature of weather, especially during winter.
Extreme events like these are essential for understanding climate phenomena. They offer insights into how local geographical features, like mountain ranges or atmospheric conditions, can influence weather patterns significantly. Such records also help researchers study climate behavior and better prepare for potential extreme weather in the future.