Newtonian Mechanics

Overview

Using Newton's Three Laws of Motion, you can begin to understand how math and physics are joined together, and how reality obeys universal and absolute laws.

Motion

You can use a Vector to describe the position of a particle in space at any given moment in time. The position vector will probably change over time, so it can be described as a vector function of time.

We like to use <math>\mathbf{r}(t) \equiv \vec r(t)\,</math>. Vectors can be in any number of dimensions. If you're using rectangular coordinates (which you likely are at this level), then you can describe the position with as many real numbers as you have dimensions. Each component of the vector is indicated as <math>\vec r = (r_1\ r_2\ r_3)^T = (r_x\ r_y\ r_z)^T\,</math>. Each of these will vary as a function over time, so really you have <math>\vec r(t) = (r_1(t)\ r_2(t)\ r_3(t))^T = (r_x(t)\ r_y(t)\ r_z(t))^T\,</math> in 3 dimensions. We can also describe the position vector with geometry. We note that position has a direction and a length.

(Note that physicists are not nearly as precise in their math. I'm adding the t-parameter simply as a formality. Most physicists assume what the parameters of a function are, and re-use the same symbol even if the function varies by different parameters.)

The position vector varies over time, and how it varies is something we are already acquainted with. The velocity is simply the first time-derivative of the position vector: <math>\vec v(t) = {d \over dt} r(t)</math>. Like the position vector, the velocity has a direction and magnitude. We call the magnitude of the velocity vector the "speed"; it's what you would read on your speedometer as you traveled in the car. The direction is what we call the "direction", when we don't clarify the direction such as "the direction of the position" or "the direction of the acceleration."

The acceleration vector varies over time as well, and is simply the second time derivative of the position, or the time derivative of the velocity. We usually use <math>\vec a(t)\,</math> for the acceleration, and note that it has a direction and magnitude as well. Generally, we are only interested in the acceleration in the direction of the velocity (which causes the speed to change), or perpendicular to the velocity (which causes the direction of the velocity to change, without affecting the speed.)

The above three concepts are fundamental to any further discussion. You should be able to comprehend a variety of motions:

• Object at rest. (<math>\vec r = \vec r(0); \vec v = \vec a = 0</math>)
• Object in motion with no acceleration. (<math>\vec a = 0; \vec v = \vec v(0); \vec r = \vec r(0) + \vec v(0) t</math>)
• Object under constant acceleration in the direction of the velocity: (<math>\vec a = \vec a(0); \vec v = \vec v(0) + \vec a(0) t; \vec r = \vec r(0) + \vec v(0) t + {1 \over 2}\vec a(0) t^2</math>)
• Object under constant acceleration in the opposite direction of the velocity (slows down, stops, accelerates backwards.)
• Object under acceleration perpendicular to the velocity (forms a parabola)
• Object under varying acceleration (more complicated to solve, but you should get a "feel" for how things move.)

Forces and Momentum

The concept of momentum is rather simple, but cannot be completely understood without force. To begin with, you must change the way you see the world. Newton's First Law basically states that things would keep on moving in the same direction, forever and ever, or stay at rest, forever and ever, if they do not have forces acting on them.

In reality, if I slide a book across a table, it slows down to a stop---acceleration in the opposite direction to is velocity. What we must now state is that the reason there was acceleration was because there were forces acting on that book.

Forces cause things to change their velocity. Forces and acceleration are related in some fundamental way. However, any given object can have multiple forces acting on it, depending on how precise we want to be. The sliding book on the table, for instance, is still feeling the force of gravity, which pulls everything down towards the earth. It also feels the force of the table pushing up on it, in exact amount so as to prevent the book from falling through the table. There is also the force of drag, that effect which we really don't feel until we are moving at fast speeds, such as down the road. If someone is pushing the book, then the fingers are putting a force on the book, much like the force the table exerts on the book to prevent the book from going through the fingers (or vice-versa!) Then there is the force of friction, the force which causes the book to "stick" to the table rather than glide off effortlessly.

Each force has a magnitude and a direction---a vector. Some point up, others down, and still others from side to side. Some are strong (gravity, friction, contact), some are weak (drag).

We're really not interested in each force independent of one another. After all, the book can only have one position, one velocity, and one acceleration. But we know that the forces "add" up, and may cancel each other out or supplement each other. So we really want to think about the sum of the forces.

Force, Mass and Acceleration

Newton's Second Law relates forces to acceleration. They are exactly proportional to one another, by the factor of the mass of the object involved.

What is mass? It is slightly different than the concept of "weight". Weight is really the force of gravity. In space, orbiting around the earth, your scale will register 0 weight. However, your mass hasn't changed at all since you were on earth.

Weight is useful, when you want to consider the force of gravity, but that's all it's good for. Mass, on the other hand, is truly constant, and rarely do we consider systems that change their mass over time. (Rocket scientists, you're special.)

Newton described how forces, mass, and acceleration relate with this simply formula you should commit to memory and sear into your being forever and ever:

<math>\sum \vec F = m \vec a</math>

In plain English, the net force acting on an object is the same as the acceleration times the mass of the object.

This works both ways; both sides are always going to be equal, for everything everywhere for all time. If you know the net force and mass, you know the acceleration. If you know the net force and acceleration, you can calculate the mass. If you know the mass and the acceleration, you know the net force!

I hope you start to see these two things merge in your mind: Force and acceleration times mass are the same thing, fundamentally.

Thinking like this, we can state force in these terms:

<math>\sum \vec F = m \vec a = m {d \over dt} \vec v = {d \over dt} m \vec v\,</math>

As happens so often in Physics, we discover some interesting relationship between physical quantities and out falls a new quantity. In this case, it is <math>{d \over dt} m \vec v\,</math>. What could that possibly be? Why, it's simply the net force.

But what about <math>m \vec v\,</math>? What does that represent? A massive velocity? Indeed, conceptually we know that things that are moving quickly are much more dangerous when they are massive. A paper airplane moving at a good speed will annoy someone, but a brick moving at the same speed can send them to the hospital.

This quantity, which we name momentum (<math>\vec p</math>), is a lot simpler to consider. In fact, for rocket scientists, who know that mass can change over time when you are expelling propellant out of the tail, they would rather think about momentum and solve for that, and then figure out what the velocity is at any given time rather than the other way around.

Solving Problems

Now, let's take our toolset and solve the problems of motion for a variety of situation. All we have to do is relate the position of the particle to its net forces using simple differential equations, and we have our answers to extraordinary precision to life's problems.

The number of problems you can solve this way should give you a feel for how powerful just a little mathematical application to the physical world can be. Keep in mind, however, that what you can do is nothing compared to what you will be able to do in the near future, as you learn a little more math and understand a different way to view the world around you.