## Section 3.2: Understanding Mass-Energy Equivalence

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*E* = *mc*^{2}. (3.3)

This is the one physics equation that almost everyone knows, but most do not understand. The animation shows a *thought experiment* that is generally attributed to Einstein as a way to see the equivalence of mass and energy.^{1} Restart.

A laser, attached to a box, emits light. The box is on a frictionless surface. The light carries with it energy and momentum: *p* = *E*/*c*. Since light has momentum, the box must *recoil*, much like a gun recoils upon firing a bullet. In the animation, therefore, we see the box move. When the light hits the right side of the box, it is absorbed, bringing the box to a stop. What happens to the center of mass? The
center of mass of the box has clearly moved to the left. Since there are no external forces, the center of mass of the system (box plus light) must remain fixed. To keep the center of mass of the system fixed, the light (which travels from left to right) must provide the missing mass. However, light has no mass, just energy. Thus, the energy of the light must be equivalent to the amount of mass needed to keep the center of mass of the system fixed.

Quantitatively, we can compare the distance the box moved with the center of mass of the light motion. Assuming the box is massive enough that its motion is non-relativistic, the amount the box moves is determined from conservation of momentum:

*p*_{box }= −*p*_{light }= −*E*/*c* ,

which implies v_{box }= E/*Mc* where *M* is the mass of the box.

When the light hits the left side of the box, and is absorbed, the box stops. The time that it takes the light to hit is *approximately* *L*/*c* (it is actually shorter because the box is moving toward the light, but for our purposes, we will neglect this small effect). So, the distance the box moves is

*d*_{box }= v_{box }*t* = *EL*/*Mc*^{2 }.

For the center of mass to stay fixed, the light, which traveled a distance *L*, must have a *mass equivalent* given by the following:

*Md*_{box} = *m*_{equiv light }*L* ,

or

*MEL*/*Mc*^{2 }= *m*_{equiv light }*L* ,

and hence, *m*_{equiv light }= *E*/*c*^{2}, the equivalence of mass and energy!

^{1}Based on the development in J. Bernstein, P. M. Fishbane, and S. Gasiorowicz, *Modern Physics,* Prentice Hall (2000), pp.81-82.

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