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When a photon interacts with a deuteron, the kinetic energy released is a key aspect of the resulting process. In this scenario, the energy of the incoming photon is converted into the kinetic energy of the resulting particles, assuming no energy is lost during the interaction.
Calculating kinetic energy in this context involves understanding that the total energy from the photon is transferred to the kinetic energy of the resulting proton and neutron. To find the kinetic energy, we use:

• Photon energy formula: this gives the total energy available for conversion into kinetic energy.

The kinetic energy released here is approximately calculated as:\[ K.E. = 5.675 \times 10^{-14} \text{ J} \]This describes the amount of kinetic energy available, perfectly shared between the neutron and the proton after the photon's interaction with the deuteron.

The photon energy can be detailed using the equation:\[ E = \frac{hc}{\lambda} \]where \( h \) is the Planck's constant \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \), \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \), and \( \lambda \) is the photon's wavelength. This equation comes from the understanding that a photon carries quantized energy, dependent on its wavelength.
This direct calculation helps us determine the energy delivered by the photon involved in the reaction. Given a wavelength of \( 3.50 \times 10^{-13} \text{ m} \), substituting these values gives the energy:\[ E \approx 5.675 \times 10^{-14} \text{ J} \]This equation is fundamental in physics, underscoring the relationship between a photon's energy and its wavelength.

To determine how fast the particles (proton and neutron) move after the deuteron splits, we use the concept of kinetic energy sharing. The kinetic energy formula is:\[ K.E. = \frac{1}{2}mv^2 \]where \( m \) is the mass, and \( v \) is the speed we need to calculate. Given that both particles equally share the total kinetic energy, each receives:\[ \text{K.E. per particle} = \frac{5.675 \times 10^{-14}}{2} = 2.8375 \times 10^{-14} \text{ J} \]Plugging this energy into the kinetic energy formula allows us to solve for \( v \), the speed of each particle. Rearranging terms and solving gives:\[ v \approx \sqrt{\frac{(2 \times 2.8375 \times 10^{-14})}{1.66 \times 10^{-27}}} \approx 2.92 \times 10^7 \text{ m/s} \]This result tells us the velocity at which each resulting particle travels, providing insight into the dynamics of particle interactions.

Converting atomic mass units (amu) to kilograms is crucial when calculating the speed of particles. One atomic mass unit (\( 1 \text{ u} \)) is approximately equal to \( 1.66 \times 10^{-27} \text{ kg} \). This conversion is necessary because the equations used in kinetic energy calculations require mass in kilograms.

• Ensure accuracy by consistently using this conversion factor in calculations involving atomic particles.

In our scenario, both the proton and the neutron are assumed to have a mass of \( 1.00 \text{ u} \), translating to a mass of approximately \( 1.66 \times 10^{-27} \text{ kg} \) each.
Knowing their mass in kilograms allows us to use the kinetic energy formula to accurately determine their speeds. Properly understanding unit conversions is foundational to solving physics problems involving atomic particles.