E=mc²: The Full Story

E equals mc squared may be the most famous equation in the world, but what you might not know is that it isn't the whole story. It just describes objects that have mass and that aren't moving. The full equation is E squared equals mc squared squared plus p times c squared, where p represents the momentum of the object in question. This might all seem a bit confusing, but in fact you can draw it as a right triangle with sides E, mc squared and p times c. And just use the Pythagorean theorem, a squared plus b squared equals c squared to give you the equation. Also, from here it's clear to see that for an object that isn't moving and thus doesn't have any momentum and thus p is zero, we get back our good old friend E equals mc squared. On the other hand, if the particle in question is massless, like light, then mass is zero, and we get E equals p times c. This tells us that the energy of a massless particle, like a photon of light, is the same as its momentum, up to a factor of the speed of light. In fact, the closer the energy of something is to p times c, the closer that something is to behaving like light. I mean, look here, this tiny little bit of mass is hardly mass at all. Anyway, as an example, an object's velocity is equal to the speed of light, times the ratio of the object's momentum to energy, or pc over E. If your momentum increases, p times c gets closer and closer to equaling your energy, so their ratio gets closer and closer to being one, and your speed gets closer and closer to light speed. But because of that tiny little bit of mass, the momentum side of the triangle will always be a little bit smaller than the energy side. No matter how hard you try to increase your momentum, it never quite gets to the point where p times c equals your energy, and thus your velocity can never quite reach the speed of light, all because the hypotenuse of a right triangle is longer than its legs.