Problem 14 - University Physics with Modern Physics

Question

Explain why the average velocity of air molecules in a closed auditorium is zero but their root-mean-square speed or average speed is not zero.

Step-by-Step Solution

Answer

Answer: The average velocity of air molecules in a closed auditorium is zero because individual velocities of the molecules in various directions cancel each other out, resulting in no net displacement. However, their root-mean-square speed and average speed are not zero because these are scalar quantities and are not affected by the direction of motion. These quantities provide information about the magnitude of motion experienced by the molecules, which remains non-zero due to their constant random motion.

Step 1: Understanding Average Velocity

Average velocity is a vector quantity that is calculated by taking the total displacement (change in position) of an object and dividing it by the total time taken to cover that displacement. If the object returns to its starting position, its average velocity is zero since there is no net displacement.

Step 2: Understanding Root-Mean-Square Speed

Root-mean-square (RMS) speed is a scalar quantity that is calculated by taking the square root of the average of the squared speeds of the individual molecules. As it's an average of the squared speeds, it represents the "typical" speed of a random molecule in a sample of gas.

Step 3: Understanding Average Speed

Average speed is a scalar quantity that is calculated by taking the total distance traveled by an object and dividing it by the total time taken to travel that distance. This gives us the measure of an object's overall rate of motion without considering its direction.

Step 4: Explaining Zero Average Velocity in a Closed Auditorium

In a closed auditorium, air molecules are in constant random motion, colliding with each other and the walls of the auditorium. These random motions in various directions cause the individual velocities of the molecules to cancel each other out. As a result, the sum of their displacements is zero, making their average velocity zero.

Step 5: Explaining Non-Zero Root-Mean-Square Speed and Average Speed

Although the average velocity of air molecules in a closed auditorium is zero, their root-mean-square speed and average speed are not zero because they are scalar quantities and are not affected by the direction of motion. These quantities give information about the magnitude of motion experienced by the molecules in a sample of gas. Since air molecules are in constant random motion, their root-mean-square speed and average speed remain non-zero even though their average velocity is zero.