## Newton’s laws of motion

LAW 1

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Newton’s first law is often called the law of inertia, for it defines bodies in inertial frames of reference, ie. with constant velocity (where the velocity could be zero). In modern parlance we’d say

$\displaystyle \sum{ \vec{F} } = 0 \Rightarrow \frac{ d\vec{v}}{dt} = 0$

In the absence of a force, an object will continue along whatever inertial trajectory its on. If you throw a dart and neglect air resistance whose effects are minute over the range of consideration, the horizontal component of its velocity remains constant since there are no forces acting on it in that line. However, it eventually hits the ground because the vertical component of force on it is non-zero, which brings use to the second law:

LAW 2

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which the force is impressed.

Newton goes on to say, in effect, that these forces obey the law of superposition and thus can be added simply. Using more modern terms and notation, we’d say that the sum of forces acting on a body has the effect of changing its momentum, such that:

$\displaystyle \sum{ \vec{ F } } = \frac{ d \vec{p}}{dt} = \frac{ d (m \vec{v})}{dt} = m \frac{ d \vec{v}}{dt} = m \vec{a}$

Note that what we call the Newtonian approximation crept in there, where we assumed that the mass $\displaystyle m$ was not dependent on the velocity.

Recently I watched the movie Wanted where the characters would whip their arms when firing their pistols which would cause the bullets to arc around objects. Newton’s second law shows the error in this: For a bullet to arc it must change its direction; which means it must change its velocity; which means it must be accelerated; which means there must be a force acting on it in the direction in which it is arcing. Since there are no such forces that could act on the bullet, it would leave the gun with a constant velocity and hit Angelina Jolie in the face.

LAW 3

To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

An oft-misunderstood law, in modern notation we’d write

$\displaystyle \vec{F}_{12} = -\vec{F}_{21}$

That is, forces always come in equal and opposite pairs. What’s important to remember is that it is the forces that are equal (and opposite), not the accelerations or velocities or whatever. When you push your hand against a wall the contact forces are equal and opposite, but since the mass of your hand is so much less than the mass of the wall, it is accelerated and so the fingers are bent back whereas the wall seems impressed not at all. When the hammer of a gun strikes the bullet, the recoil kick that the shooter feels is equal in magnitude to what the bullet feels, but it has a small mass and so experiences a much higher acceleration and so goes soaring. When you sit in a chair and pull up on the arm rests, you can never lift yourself off the floor since the force the armrests impart to your hands are exactly compensated by an equal force that you exert down on the seat (in other words, a body cannot exert a net force upon itself, so if you’re ever stuck floating in space with nothing to throw, you’re hooped).

Given the age Newton lived in, you can see how the laws were expressed with some verbosity when more modern mathematical notation can compress the information and offer certain enlightenment. For reasons like that, I’ve come to consider the notations used in math and science to be a form of technology, a set of tools that we have available that our ancestors did not. I was reading today about general relativity books, and how the older texts used “outdated notation,” which may not sound like much of a thing at first, but its similar in a sense to trying to learn mechanics by reading Newton. While we thank the man for his contributions to human civilization and read him later for intellectual satisfaction, you’d be pointlessly limiting yourself by attempting to adopt the limitations he himself was burdened with by an accident of time and history.