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Kinetic energy is the energy that an object possesses due to its motion. It depends on both the mass of an object and the square of its velocity.
The formula for kinetic energy, specifically for objects moving at non-relativistic speeds (speeds much less than light speed), is given by: \[ KE = \frac{1}{2} mv^2 \] where:

• \( KE \) is the kinetic energy,
• \( m \) stand for the mass of the object,
• \( v \) is the velocity (in this scenario, \( v = 0.05c \) where \( c \) is the speed of light).

When a particle moves, it uses some of its internal energy to move, which we can measure using the above kinetic energy formula. For the particle in the exercise, the kinetic energy is equated to the photon energy to find the desired wavelength ratios.

Photon energy pertains to the energy carried by a single photon, the basic unit of light and all other forms of electromagnetic radiation. It is an essential concept in understanding quantum physics. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. The formula is given by: \[ E_{photon} = \frac{hc}{\lambda_{photon}} \] where:

• \( E_{photon} \) stands for photon energy,
• \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js),
• \( c \) is the speed of light in a vacuum (approximately \( 3.00 \times 10^8 \) m/s),
• \( \lambda_{photon} \) is the wavelength of the photon.

This formula helps bridge the classical and quantum worlds by linking the wavelength of light to its energy. Understanding photon energy is crucial in technologies like solar power, lasers, and various quantum mechanisms.

The wavelength ratio is a comparative measure between two different types of wavelengths. In our scenario, this involves comparing the photon's wavelength with the de Broglie wavelength of the particle. Each of these wavelengths is determined by separate principles, but their comparison is instrumental for unraveling quantum mechanical phenomena. The photon's wavelength, based on its energy, is determined using the photon energy formula: \[ \lambda_{photon} = \frac{2hc}{mv^2} \] The de Broglie wavelength for a particle is derived from its momentum: \[ \lambda_{dB} = \frac{h}{mv} \] To find their ratio, the expression becomes: \[ \frac{\lambda_{photon}}{\lambda_{dB}} = \frac{2v}{v} = 0.1 \] This result shows that the photon's wavelength is a tenth of the de Broglie wavelength, given the conditions of the movement speed of the particle. Wavelength ratios like these are critical in understanding how particles behave like waves, fundamental in the world of quantum mechanics.