# A particle has a kinetic energy of 62 MeV and a momentum of 335 MeV/c. Find its mass (in MeV/c^2)...

## Question:

A particle has a kinetic energy of 62 MeV and a momentum of 335 MeV/c. Find its mass (in MeV/c{eq}^2 {/eq}) and speed (as a fraction of c).

## Kinetic Energy:

The value of the energy attained in the body due to the motion or movement is called kinetic energy. The mathematic formula used to determine the magnitude of the kinetic energy is {eq}K.E = \dfrac{1}{2}m{v^2}. {/eq}

Given data

• The value of the kinetic energy of the particle is {eq}K.E. = 62\;{\rm{MeV}} {/eq}
• The value of the momentum of the particle is {eq}P = 335\;{\rm{MeV/c}} {/eq}

The expression for the relation between the kinetic energy and the momentum is:

{eq}\begin{align*} P &= \sqrt {K.E\left( {K.E + 2{m_o}{c^2}} \right)} \\ {P^2} &= K.E\left( {K.E + 2{m_o}{c^2}} \right) \end{align*} {/eq}

Substitute the values in the above equation.

{eq}\begin{align*} {\left( {335} \right)^2} &= 62\left( {62 + 2{m_o}{{\left( c \right)}^2}} \right)\\ {m_o}{\left( c \right)^2} &= 874.040\;{\rm{MeV}}\\ {m_o}{\left( c \right)^2} &= \left( {874.040 \times 1.6 \times {{10}^{ - 13}}} \right)\;{\rm{kg}}\\ {m_o}{\left( {3 \times {{10}^8}} \right)^2} &= \left( {874.040 \times 1.6 \times {{10}^{ - 13}}} \right)\;{\rm{kg}}\\ {m_o} &= 155.3848 \times {10^{ - 29}}\;{\rm{kg}} \end{align*} {/eq}

Thus, the value of the mass of the particle is {eq}155.3848 \times {10^{ - 29}}\;{\rm{kg}} {/eq}

The expression for the velocity of the particle in comparison with the velocity of light is:

{eq}P = \dfrac{{{m_o}v}}{{\left( {\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} } \right)}} {/eq}

Substitute the values in the above equation.

{eq}\begin{align*} 335 &= \dfrac{{\left( {874.040} \right)v}}{{\left( {\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} } \right)}}\\ 335\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} &= 874.040v\\ 112225\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right) &= 763945.9216{v^2}\\ 112225{c^2} - 112225{v^2} &= 763945.9216{v^2}{c^2}\\ v &= 0.36c \end{align*} {/eq}

Thus, the value of the velocity of the particle is {eq}0.36c {/eq}, where {eq}c {/eq} is the velocity of light.