Equation of the Day #7: E=mc^2 and the Pythagorean Theorem

I doubt I’m the first person to introduce you to either of these equations, but if I am, then you’re in for a treat! The first equation is courtesy of Albert Einstein,

$E=mc^2.$

Just bask in that simplicity. The constant $c$ is the speed of light, which is a rather large number. In fact, light takes just over a second to travel to Earth from the moon, while the Apollo missions took three days. The fastest we’ve ever sent anything toward the moon was the New Horizons mission to Pluto, and it passed the moon after a little over eight-and-a-half hours. So, light is pretty fast, and therefore c2 is a gigantic number. The E and m in this equation are energy and mass, respectively. If we ignore the c2, which acts as a conversion rate, E=mc2 says that energy and mass are equivalent, and that things with mass have energy as a result of that mass, regardless of what they’re doing or where they are in the universe. This equation is at the heart of radioactive decay, matter/antimatter annihilation, and the processes occurring at the center of the sun. Now, that all may sound foreign, but it’s at work constantly, and we take advantage of it. For instance, positron emission tomography scans, or PET scans, are used to image the inside of the body using a radioactive substance that emits positrons (the antimatter counterpart to the electron). These positrons annihilate with the electrons in your body and release light, which is then detected by a special camera. This information is then used to reconstruct an image of your insides.

As numerous as the phenomena are that E=mc2 covers, it’s actually not the full story. After all, objects have energy that isn’t a result of their mass; they can have energy of motion (kinetic energy) and energy due to where they are in the universe (potential energy). Ignoring potential energy for the moment, you may be wondering how to include the kinetic energy in our simple equation. As it turns out, special relativity says that the energy contributions obey the Pythagorean theorem,

$a^2 + b^2 = h^2,$

which relates the three sides of a triangle whose largest angle is 90°.

(I’ve called the longest side h instead of c to avoid confusion with the speed of light.) In our example, the total energy E is the longest side, and the “legs” are the rest energy (mc2) and the energy contribution due to momentum, written as pc.

In equation form,

$E^2 = (pc)^2 + (mc^2)^2.$

This equation only holds for objects moving at constant velocity, but the geometric relationship between the total energy and its contributions is very simple to grasp when shown as a right triangle. In fact, this isn’t the only place in special relativity where Pythagorean relations pop up! The formulas for time dilation and length contraction, two phenomena that pop up in special relativity, are governed by a similar equation. For instance, time dilation follows the triangle

where tmov is the time elapsed according to a moving clock, trest is the time read by a clock at rest, and s is the distance that the moving clock has covered over the elapsed time. How can two clocks read different times? I’ll save that question for another day.

-A

Equation of the Day #6: Newton’s Second Law

Born on Christmas Day, 1642, Isaac Newton was one of the most brilliant minds the world has ever seen. He invented calculus, discovered that white light is composed of all the colors of the rainbow, formulated the law of universal gravitation (which we now use to send people and robots into orbit around Earth and other objects in the Solar System), and his famous laws of motion (among a host of other achievements). Today I’m focusing on his Second Law of Motion, which you’ve probably seen written as

${\bf F}=m {\bf a},$

where F is force, m is mass, and a is acceleration. The boldface formatting indicates a vector, which is a mathematical entity that has both magnitude (or “strength”) and direction (up/down, left/right, forward/backward). What this means is that an object whose speed (the magnitude of velocity) or direction of travel is changing must be subject to an unbalanced force. This ties in to Newton’s First Law of Motion, which states that the natural state of an object is to move in a straight line at constant speed. However, the above equation is actually not fully general – it only applies when the accelerating object has constant mass. This works fine for projectile motion of a pitcher hurling a baseball or a cannonball launched at an enemy fortress, but it fails for, say, a rocket delivering a payload to the International Space Station, since fuel is being burned and ejected from the rocket. In general, Newton’s Second Law is given by

${\bf F}=\dfrac{d{\bf p}}{dt},$

where p is the total momentum of the system. Momentum is a measurement of the “oomf” that an object has due to its motion. Explicitly, it is the product of mass and velocity, or ${\bf p} = m{\bf v}$. The expression d/dt indicates the rate of change of a quantity, in this case momentum, over time. So Newton’s Second Law states that a change in an object’s motion is due to an unbalanced force, which sounds like what I said for the equation ${\bf F}=m {\bf a},$ but this takes into account a change in mass as well.

Conceptually, this was a big breakthrough at the time and is something that students in introductory physics classes struggle with today. (Note: Newton figured all this stuff out and more by the time he was 26. Let that sink in.) On a quantitative level, this law allows us to predict the motions of objects that subjected to certain kinds of forces. If we know the nature of the force an object is subjected to, we can make predictions of what path the object will travel along. Similarly, if we know the trajectory of an object, we can predict the behavior of the force acting upon it. This was how Newton figured out the nature of gravity. Observations of his time showed that, from a sun-centered view of the Solar System, the planets orbit in ellipses, not circles. Newton surmised that this meant that the force of gravity must diminish with the square of the separation between two objects.

But how? The presence of acceleration in Newton’s Second Law indicates a differential equation. Acceleration is the rate of change of velocity, which itself is the rate of change of position (which I will denote as r). So, taking that into account, we get the equation

$\dfrac{d^2{\bf r}}{dt^2} = \dfrac{{\bf F}}{m}$

This can be solved for an expression of position as a function of time when we know the form that F takes. If the force is constant in space (like gravity near Earth’s surface), r takes on the form of parabolic motion. If the force obeys Hooke’s Law, $F = -k \Delta r$, then the solution is simple harmonic oscillation. And if the force obeys an inverse square law, the position takes the functional form of conic sections, which are ellipses for the planets bound to the Sun in our solar system.

Conic sections: 1. Parabola, 2. Circle (bottom) and ellipse (top), 3. Hyperbola. (Image: Wikimedia Commons)

This is the model we use when we send objects into orbit or when we want to model a baseball being hit out of the park (among many other things), and it has been extended to explain phenomena that Newton himself could not have imagined. When scientists or mathematicians describe equations as beautiful, this pops into my head. It has such a simple form, but explains a wealth of real phenomena.