The energy $$ E $$ and momentum $$ p $$ of a moving body of mass $$ m $$ are related by some equation. Given that c represents the speed of light, identify the correct equation

We need to identify the correct relativistic energy-momentum relation.

Einstein's Energy-Momentum Relation:

In special relativity, the total energy $$E$$ of a particle with rest mass $$m$$ and momentum $$p$$ is given by:

$$E^2 = (pc)^2 + (mc^2)^2 = p^2c^2 + m^2c^4$$

Derivation:

The relativistic energy is $$E = \gamma mc^2$$ and relativistic momentum is $$p = \gamma mv$$, where $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$.

$$E^2 - p^2c^2 = \gamma^2 m^2 c^4 - \gamma^2 m^2 v^2 c^2 = \gamma^2 m^2 c^2(c^2 - v^2) = m^2c^4$$

(since $$\gamma^2(c^2-v^2) = c^2$$)

Therefore: $$E^2 = p^2c^2 + m^2c^4$$.

The correct answer is Option 4: $$E^2 = p^2c^2 + m^2c^4$$.