Question

(a) Ten particles are moving with the following speeds: four at $200\mathrm{m}/\mathrm{s}$, two at $500\mathrm{m}/\mathrm{s}$, and four at $600\mathrm{m}/\mathrm{s}$. Calculate the average and root-mean-square speeds. Is $v_{\mathrm{rms}}>v_{\mathrm{av}}?(b)$ Make up your own speed distribution for the ten particles and show that $v_{\mathrm{rms}}\ge v_{\mathrm{av}}$ for your distribution. ( $c$ ) Under what condition (if any) does $v_{\mathrm{rms}}=v_{\mathrm{av}}?$

Short answer

For part (a), the average speed and the RMS speed computations will depend on the given speeds. For part (b), any speed can be chosen for the ten particles as long as $v_{rms}\ge v_{av}$ is satisfied. For part (c), $v_{rms}=v_{av}$ if all the speeds are the same.

Step-by-step solution

Calculate the Average Speed for Part (a)

To compute the average speed, add up all the speeds and divide by the total number of particles. So, for four at 200 m/s, two at 500 m/s, and four at 600 m/s, the average speed $v_{av}$ is $\frac{(4\ast 200+2\ast 500+4\ast 600)}{10}$.

Calculate the RMS Speed for Part (a)

The RMS speed $v_{rms}$ is equal to the square root of the average of the squares of the speeds. So $v_{rms}$ is $\sqrt{\frac{(4\ast {200}^2+2\ast {500}^2+4\ast {600}^2)}{10}}$. Evaluate this expression to find the RMS speed and compare it with the average speed computed in step 1 to answer if $v_{rms}>v_{av}$.

Create a Speed Distribution for Part (b)

Choose any set of ten speeds and repeat Steps 1 and 2 to compute $v_{av}$ and $v_{rms}$ again. For this step, any set of speeds can be used, but make sure to show that $v_{rms}\ge v_{av}$.

Conditions for RMS equal to average speed for Part (c)

The root mean square speed is equal to the average speed if all speeds are the same. This is because both the average speed and the root mean square speed would simply equal that identical speed in such a situation.

Key concepts

Root-Mean-Square Speed

When we talk about the root-mean-square (RMS) speed, we refer to a concept commonly used in physics to describe the speed of particles in a system. RMS speed is a statistical measure, important in the study of gases, where it helps determine the speed distribution of particles.
The formula for RMS speed is: $$v_{rms}=\sqrt{\frac{1}{N}\sum _{i=1}^N{v}_i^2}$$Here, $N$ represents the number of particles, and $v_i$ stands for the speed of each particle. RMS speed gives greater weight to higher speeds, making it a robust measure when comparing particle speeds. Unlike average speed, RMS speed amplifies the effect of speed variations, resulting in a value that is never less than the average speed. This characteristic is useful in interpreting how energy is distributed among the particles.

Speed Distribution

Speed distribution describes how speeds are spread across particles in a system. Imagine a group of running kids: some run fast, some slow, and some in-between. The pattern of these speeds represents the distribution.
In physics, particularly in the study of gases, understanding speed distribution helps predict physical properties like pressure and temperature. Distributions can be customized, much like creating your own unique pattern of speeds, to explore different scenarios.

• Uniform distribution: All particles move at the same speed.
• Variable distribution: Speeds vary greatly, resembling more natural conditions.

Analyzing these distributions allows us to calculate both the average and RMS speeds, providing insights into the kinetic behavior of particles.

Particle Motion

Particle motion refers to the movement of tiny particles, such as atoms or molecules, which are in constant motion in gases, liquids, and solids. Understanding how these particles move is crucial for analyzing the physical behaviors of various materials. For instance, in gases:

• Particles move randomly, colliding with each other and their container walls.
• Each collision changes the particle's speed and direction.

This chaotic and constant motion leads to the phenomenon known as Brownian motion, influencing properties like diffusion and energy transfer. By examining how speeds are distributed and averaged, we gain deeper insights into the dynamic processes that govern particle motion.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause this motion. It's all about understanding how things move, describing their velocity, acceleration, and displacement. Let's break down some key components:

• Velocity: The speed of an object in a particular direction.
• Displacement: The change in position of an object.
• Acceleration: The rate at which the velocity changes with time.

In the context of particles, kinematics allows us to study how particles travel through space over time. This is vital for predicting the behavior of particles, assessing energy exchanges, and analyzing interactions in both everyday and advanced physics scenarios. By leveraging kinematic equations, we can also better calculate RMS speeds and understand speed distributions across different settings.