## Section 3.3: Understanding the Energy-Momentum Equation

Please wait for the animation to completely load.

The animation shows the relationship between total energy, momentum, and mass **(energy is in MeV, mass in MeV/ c^{2} and momentum in MeV/c)**. Restart. The slider controls the velocity of the object. Set the particle mass and total energy and move the slider. The total energy,

*E*, is given by the equation

*E*^{2 }= (*pc*)^{2 }+ (*m*_{0}*c*^{2})^{2} = *p*^{2}*c*^{2 }+ *m*_{0}^{2}*c*^{4}, (3.4)

where *p* is the relativistic momentum, *c* is the speed of light, and *m*_{0} is the rest mass of the particle (the mass of the particle measured in its reference frame). In the diagram, the hypotenuse represents the maximum energy while the two legs are the mass and maximum momentum. Using this relationship, and the conservation of total energy and momentum, you can determine masses of unknown particles. In this case, the physics of supercollider experiments.

Not surprisingly, as you increase the particle speed, more of the total energy is kinetic energy (the red arrow) since

*E* = *m*_{0}*c*^{2 }+ *K*. (3.5)

Try setting the values so you have a particle with small mass and large maximum energy. What happens? Try the converse. If you have small enough mass and high enough energy (at speeds very close to *c*), you can use the extreme relativistic approximation of *E *≈* pc*.

Although *E* and *pc* vary from reference frame to reference frame (as does the kinetic energy, *K*), the quantity *E*^{2 }− *p*^{2}*c*^{2 }is relativistically invariant. This is sometimes convenient in problem solving as it can enable you to switch more easily between the laboratory reference frame to a reference frame where a particle is at rest.

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