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Kinetic energy is the energy an object possesses due to its motion. It's a fundamental concept in physics that helps us understand how motion relates to energy. For linear motion, the kinetic energy (KE) of an object is determined by the formula:\[KE = \frac{1}{2}mv^2\]where:

• \(m\) represents the mass of the object
• \(v\) represents the velocity of the object

Kinetic energy depends on both mass and velocity, but it's particularly sensitive to changes in velocity because it is squared in the equation. This means a small increase in velocity can result in a significant increase in kinetic energy. In the context of the given problem, doubling the kinetic energy affects the velocity, influencing other properties such as the de-Broglie wavelength.

In physics, a free electron is an electron that is not bound to an atom or molecule and can move freely through a conductor. Unlike electrons in atomic orbitals, free electrons have greater flexibility in movement, allowing them to conduct electricity. Free electrons play a significant role in the concept of de-Broglie wavelength because they interact with their surrounding environment, such as electromagnetic fields, and exhibit wave-particle duality. This duality is a fundamental aspect of quantum mechanics, showing that particles like electrons can behave like waves under certain conditions. When considering a free electron, changing its kinetic energy leads to changes in its velocity, hence altering its de-Broglie wavelength.

The de-Broglie wavelength describes the wave-like behavior of particles and is given by the expression:\[\lambda = \frac{h}{mv}\]where:

• \(\lambda\) is the wavelength
• \(h\) is Planck's constant
• \(m\) is the mass
• \(v\) is the velocity

When kinetic energy of a free electron is doubled, its velocity also changes, leading to a modification in the de-Broglie wavelength. The original problem asks us to determine how this doubling of kinetic energy changes the wavelength. The formula shows that the de-Broglie wavelength is inversely proportional to velocity, meaning as velocity increases, the wavelength decreases. The solution demonstrates how this relationship leads to a specific change in the wavelength when kinetic energy changes.

The relationship between velocity and kinetic energy is key to understanding changes in the de-Broglie wavelength. From the kinetic energy formula \( KE = \frac{1}{2}mv^2 \), we can express velocity in terms of kinetic energy:\[v = \sqrt{\frac{2KE}{m}}\]When kinetic energy is doubled, this relationship shows the effect on velocity. If the initial kinetic energy is \(K_1\) and the final kinetic energy is \(2K_1\), then the new velocity can be calculated as:\[v' = \sqrt{\frac{2 \times 2K_1}{m}} = \sqrt{2} \times v\]Consequently, the velocity increases by a factor of \(\sqrt{2}\), not doubled, because it's the square root that increases. Hence, when calculating the change in de-Broglie wavelength using this new velocity, the wavelength becomes \(\frac{1}{\sqrt{2}}\) times the original wavelength, showing how doubling kinetic energy results in a specific change in velocity and a corresponding change in wavelength.