# A particle and its anti-particle are directed toward each other, each with rest energy 1,000 MeV....

## Question:

A particle and its anti-particle are directed toward each other, each with rest energy 1,000 MeV. We want to create a new particle with rest energy 10,000 MeV and total energy 100,000 MeV. What must the speed of the particle and antiparticle be before the collision?

## Energy conservation principle:

When the initial energy of the system is the same as the final energy during the system performs any process, this concept is known as the energy conservation principle. This concept is used for the determination of energy losses during the process.

Given data

• Initial energy of particle is, {eq}{E_i} = 1000\;{\rm{MeV}} {/eq} .
• Final energy is, {eq}{E_f} = 10000\;{\rm{MeV}} {/eq} .
• Total required energy is, {eq}{E_t} = 100000\;{\rm{MeV}} {/eq} .

The expression for conservation of energy is,

{eq}\dfrac{{{E_1}}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }} + \dfrac{{{E_2}}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }} = {E_f}.......\left( 1 \right) {/eq}

Here, v is the speed of particles, {eq}{E_1} {/eq} and {eq}{E_2} {/eq} are the energies of particles and c is the speed of light.

According to question,

{eq}{E_1} = {E_2} = {E_i} {/eq}

Substitute the values in above expression.

{eq}\begin{align*} \dfrac{{2 \times 1000}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }} &= 10000\\ \sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} &= \dfrac{1}{5}\\ v &= 0.9 \times 3 \times {10^8}\\ v &= 2.7 \times {10^8}\;{\rm{m/s}} \end{align*} {/eq}

Thus, the speed of particle is {eq}2.7 \times {10^8}\;{\rm{m/s}} {/eq} .