# Proving Velocity Averaging: <u> = 0, <u^2> = 1/3 <v^2> & <|u|> = 1/2 <v>

• bon
In summary, the conversation discusses the behavior of molecules in a gas and the relationship between their velocities, speeds, and angles of movement. The number of molecules with specific velocities and angles can be calculated using a given formula. The question then asks to show certain relationships through integration, specifically that the average of u (a cartesian component of v) is equal to 0, the average of u^2 is equal to 1/3 of v^2, and the average of the absolute value of u is equal to 1/2 of v. Finally, the question poses the idea of taking u as the z-component of v without losing any generality and asks to express u in terms of v and X and average over v and X
bon

## Homework Statement

The molecules in a gas travel with different velocities. A particular molecule will have velocity v and speed v=|v| and will move at an angle X to some chosen fixed axis. The number of molecules in a gas with speeds between v and v + dv and moving at angles between X and X+dX to any chosen axis is given by

1/2 n f(v)dv sinX dX

Where n is the numberof molecules per unit volume and f(v) is some function of v only.

Show by integration that:

(a) <u> = 0
(b) <u^2> = 1/3 <v^2>
(c) <|u|> = 1/2 <v>

where u is anyone cartesian component of v i.e vx, vy or vz

The question says that we can take u as the z-component of v without loss of generality. Why is this? Could we equally well have taken it as the x or y-component. It then says express u in terms of v and X and average over v and X.

## The Attempt at a Solution

Not sure how I am meant to do this, so any help would be great - thanks!

hello people??

## 1. What is the significance of proving velocity averaging?

The concept of velocity averaging is important in understanding the motion and behavior of particles in a gas or fluid. It allows us to calculate the average velocity of a large number of particles and make predictions about their overall movement.

## 2. How is the equation = 0 related to velocity averaging?

The equation = 0 means that the average velocity of the particles is zero. This indicates that the particles are moving in random directions, cancelling out each other's velocity, and resulting in a net average velocity of zero.

## 3. Why is = 1/3 important in proving velocity averaging?

This equation shows that the average squared velocity of the particles is equal to one-third of the squared speed of the particles. This is a key aspect of proving velocity averaging as it demonstrates the relationship between the average velocity and the individual particle speeds.

## 4. What does <|u|> = 1/2 tell us about velocity averaging?

This equation means that the average magnitude of the velocity is equal to half of the average speed of the particles. This is another important aspect of velocity averaging as it allows us to calculate the average speed of the particles based on their average velocity.

## 5. How does proving velocity averaging help us in practical applications?

Proving velocity averaging allows us to better understand and predict the behavior of particles in various systems, such as gas and fluid dynamics. This knowledge can be applied in fields such as engineering and environmental science to improve the design and efficiency of systems and processes.

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