Friday, December 3, 2010

Newton's Law of Motion

Contrary to popular belief, we can teach students that Newton had only a single law of motion. That law is F=ma. When these four simple symbols are properly defined, that's all a physicist needs to know about Newton's laws of motion.

Textbook after textbook, website after website fails to cut to the chase and sum up Newton's three laws of motion into one simpler statement, and as a teacher I find it very frustrating. It bothers me that millions of students around the world are expected to have word-for-word understanding perfect memorization of Newton's three laws of motion as spelled out by the publisher of their school's physics textbook. The reason this bothers me is that science in general (and physics in particular) is generally about finding the simplest expression for the underlying principles involved in any phenomenon. Sure, there are details, and we do care about them. I am not arguing otherwise. But the fact is that Newton's laws can be summed up in four simple symbols, and I can't see any reason not to do so for our students.

For those physics historians who really want to memorize the words, let's look at what Isaac Newton actually wrote. According to my notes (cribbed from the book On the Shoulders of Giants, in which Stephen Hawking both transcribes and interprets several great works of science, including Newton's Principia), Newton's choice of words was:
  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.
  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 that force is impressed.
  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.
If a teacher is going to insist that students memorize all three laws and regurgitate them on command, I'd at least recommend historical accuracy. But if you want to simplify, then I'll tell you how to do it.

First, we recognize that #2 above means F=ma, where "F" stands for the net force acting on an object (and both "m" and "a" are defined below). A force is often defined as simply a push or a pull, and that works for most conversations, but when teaching this topic at middle school or above I think we should define a force as an interaction between two objects that would tend to accelerate the objects. The term "net force" simply means that forces can combine, add together, and/or cancel one another out. The forces acting on an object need to be summed with vector addition before F=ma is used.

To continue, "m" stands for mass, which is a way to quantify inertia. Inertia is an object's tendency not to change it's state of motion. Things don't speed up, slow down, or change direction without something making them do so. Some things are more obstinate than others in this regard, and mass gives us a way to measure this quality in any object.

Lastly we note that "a" stands for acceleration, which is the rate of change of velocity. Acceleration is how quickly an object is speeding up, slowing down, or changing direction. Like force, acceleration is a vector quantity, which means that the quantity has a direction. Specifically, the acceleration of an object is in the exact same direction as the net force acting on the object. "F" and "a" are in bold font because that's shorthand to note that they are vectors.

These three paragraphs to describe the equation F=ma are covered by every physics teacher and physical science teacher around the world. That's certainly not my complaint. My point is that Newton's First Law is unnecessary if you actually understand the equation for Newton's Second Law. The First Law is just telling us that when the left side of F=ma is zero, so is the right. In Newton's day, this was a major insight--he was basically pointing out to people that Galileo was correct about inertia, and Aristotle was not. I find that my students are unfamiliar enough with Aristotle that I don't need to address the misconception as a separate lesson--it's just part of the Second Law lesson.

Having done away with the need for the First Law as a separate lesson, allow me to point out that the Third Law doesn't need a separate lesson, either. It is covered by our definitions. Since a force is an interaction between two objects that would tend to accelerate the objects, I need only remind students that the Second Law will apply to both objects experiencing the force in question. Done. Again, in Newton's day it was necessary to state this as a separate concept, because action at a distance (without contact, as in gravitational forces or electromagnetic forces) was not well understood. As with the First Law, the Third addresses a misconception that my students simply don't have.

To summarize, Isaac Newton gave us a single law of motion that works helps us to predict the motion of any object we might study in AP Physics and equivalent college courses. This Law does fail to explain quantum mechanics and relativity, but it has us covered up until those topics, both of which are beyond the scope of our course and our everyday lives. For any problem we have, F=ma, and that's all there is to memorize.


  1. Tremendous! I've been teaching my students that Newton's 1st Law is redundant, but hadn't made the connection to define a force as an interaction between two objects... which therefore allows you to pull in Newton's 3rd Law!

  2. I wish I had read this before the Newton's Laws test!

  3. The first law defines the range of validity of the second:
    1st Law: There exist reference frames, which we call inertial reference frames, such that an object with no net force will move at a constant velocity.
    2nd Law: In an inertial reference frame, F=dp/dt=ma.
    3rd Law; For objects 1 and 2, if 1 exerts a force F_21 on object 2, then object 2 exerts a force F12=-F21 on object 1.

    The first law declares the existence of an infinite amount of inertial frames in which the second law is valid. Without the first law, the meaning of the equation ma=F is unclear. What is F?
    You say you can discard the first law and keep only the second and you take forces to mean "an interaction between two objects". Assume you are in a reference frame in which there is body at rest. If you change to an accelerating reference frame, the body will start accelerating. According to your understanding of the second law (i.e. valid in every reference frame), there is a force acting on the object. If force is interaction between two objects, what is the second object interacting with our body?

  4. This is the fundamental equation of classical mechanics, I agree. I do think a better way to write it, however, is from the point of the view of the object. I always tell my students we care about movement (mechanics), so we should put ourselves in the object itself. This is the basis for relative motion, frames of reference, free body diagrams, vectors: everything. So which component of the 2nd Law covers movement? The acceleration. I have my students learn the formula as a=F/m. This shows them (1) what is really moving, (2) what's making it change the way it moves, the force in the numerator, and (3) what's resisting its change in motion, mass/inertia. I find that this helps with the concept and usefulness of the formula.