**Understanding Kinetic Energy and Speed Through de Broglie Wavelengths***Problem Statement:*An electron and a neutron have the same de Broglie wavelength. Which one of the following statements regarding this situation is true?*Options:*a. The electron has more kinetic energy and a higher speed.b. The electron has less kinetic energy but a higher speed.c. The electron has less kinetic energy and a lower speed.d. The electron and the neutron have the same kinetic energy but the electron has a higher speed.e. The neutron has more kinetic energy but the two have the same speed.*Analysis:*To understand the problem, let's start by recalling the de Broglie wavelength formula:\[ \lambda = \frac{h}{mv} \]where $\lambda$ is the wavelength, $h$ is the Planck constant, $m$ is the mass of the particle, and $v$ is its velocity.Given that both particles (electron and neutron) have the same de Broglie wavelength, we get:\[ \lambda_\text{electron} = \lambda_\text{neutron} \]\[ \frac{h}{m_\text{electron} v_\text{electron}} = \frac{h}{m_\text{neutron} v_\text{neutron}} \]This simplifies to:\[ m_\text{electron} v_\text{electron} = m_\text{neutron} v_\text{neutron} \]From this, we can infer:- The momentum ($mv$) of both particles is the same.The kinetic energy ($\text{KE}$) is given by:\[ \text{KE} = \frac{1}{2} mv^2 \]Using the above relationship:\[ v = \frac{p}{m} \]For particles of different masses:- Given that the mass of a neutron ($m_\text{neutron}$) is significantly greater than the mass of an electron ($m_\text{electron}$), the electron must have a higher velocity ($v_\text{electron}$) to compensate.Finally, comparing kinetic energy:\[ \text{KE}_\text{electron} = \frac{1}{2} m_\text{electron} v_\text{electron}^2 \]\[ \text{KE}_\text{neutron}

let mass of electron = memass of neutron = mn

An electron and a statements regarding this situation is true? l=h₂ a The electron has more kinetic energy and a higher speed. b. The electron has less kinetic energy but a higher speed. The electron has less kinetic energy and a lower speed. C. The electron and the neutron have the same kinetic energy but the electron has a higher speed. e. The neutron has more kinetic energy but the two have the same speed. ch (U=

An electron and a statements regarding this situation is true? l=h₂ a The electron has more kinetic energy and a higher speed. b. The electron has less kinetic energy but a higher speed. The electron has less kinetic energy and a lower speed. C. The electron and the neutron have the same kinetic energy but the electron has a higher speed. e. The neutron has more kinetic energy but the two have the same speed. ch (U=

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Relativistic Energy and Momentum

According to classical physics, time flow is the same for every observer. But according to einstein's theory, time flow is different for different observers. The idea that time flow is not the same for all may sound strange because we may not experience this effect in everyday life. This effect is only observed when an object moves with some velocity comparable to the velocity of light. This velocity is termed relativistic velocity.

Question

**Understanding Kinetic Energy and Speed Through de Broglie Wavelengths**

*Problem Statement:*
An electron and a neutron have the same de Broglie wavelength. Which one of the following statements regarding this situation is true?

*Options:*
a. The electron has more kinetic energy and a higher speed.
b. The electron has less kinetic energy but a higher speed.
c. The electron has less kinetic energy and a lower speed.
d. The electron and the neutron have the same kinetic energy but the electron has a higher speed.
e. The neutron has more kinetic energy but the two have the same speed.

*Analysis:*
To understand the problem, let's start by recalling the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
where $\lambda$ is the wavelength, $h$ is the Planck constant, $m$ is the mass of the particle, and $v$ is its velocity.

Given that both particles (electron and neutron) have the same de Broglie wavelength, we get:
\[ \lambda_\text{electron} = \lambda_\text{neutron} \]
\[ \frac{h}{m_\text{electron} v_\text{electron}} = \frac{h}{m_\text{neutron} v_\text{neutron}} \]

This simplifies to:
\[ m_\text{electron} v_\text{electron} = m_\text{neutron} v_\text{neutron} \]

From this, we can infer:
- The momentum ($mv$) of both particles is the same.

The kinetic energy ($\text{KE}$) is given by:
\[ \text{KE} = \frac{1}{2} mv^2 \]

Using the above relationship:
\[ v = \frac{p}{m} \]

For particles of different masses:
- Given that the mass of a neutron ($m_\text{neutron}$) is significantly greater than the mass of an electron ($m_\text{electron}$), the electron must have a higher velocity ($v_\text{electron}$) to compensate.

Finally, comparing kinetic energy:
\[ \text{KE}_\text{electron} = \frac{1}{2} m_\text{electron} v_\text{electron}^2 \]
\[ \text{KE}_\text{neutron}

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