Displacement: A vector of physical displacement between two points in space. Displacement has direction. This typically uses the algebraic symbol "s" or sometimes "d". In our everyday lives, we express this in units of inches, feet, centimeters, meters, kilometers, miles.
Distance: Magnitude of Displacement vector. Distance has no direction, only a number. It is often misused to indicate displacement.
Velocity: A vector of change in distance over time. Velocity has direction. This typically uses the algebraic symbol "v". In our everyday lives, we typically express this in mph (miles per hour), or kph (km per hr), or meters per second (m/s).
Speed: Magnitude of a Velocity vector. Speed has no direction, but the term is often misused to indicate velocity.
Acceleration: A vector of change in velocity over time. Acceleration has direction. This typically uses the algebraic symbol "a". In our everyday lives we don't usually use these units, but they are usually expressed in units of meters/sec^2 (m/s^2).
Jerk: A vector of change in acceleration over time. Jerk has direction. Impulse denotes the "quality" of an acceleration. For instance, the hard braking of a car has a high jerk because it imparts an acceleration on the car very quickly. Please don't get this confused with acceleration itself. This is an even less commonly used term/quantity than acceleration, and is usually denoted in units of m/s^3
Newton's Three Laws of Motion:
1) The Law of Inertia. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. In other words, a body at rest stays at rest; a body in motion stays in motion. This says that in absence of any external forces, a body will move at a constant velocity. example: an astronaut drifting in space with no propulsion will keep moving at a constant velocity until an external force is applied. He will not be able to even swim in space because there is no fluid to push against.
2) F = m*a In layman's terms, this defines Force as the acceleration of a given mass. More technically, F = dP/dt, where P = momentum. When the mass stays constant, then F = m*dv/dt = m*a. Where m = mass, v = velocity, dv/dt = derivative of velocity over time, a = acceleration. Note, that in the case of cars and rocket ships, the mass of the body is changing due to burning of fuel, so it is no longer simply mass times acceleration. This law defines Force.
3) The Law of Action-Reaction. For every action there is an equal and opposite reaction. This says that if you exert a force on an object, an equal and opposite reaction force must exist. In the case of an everyday interaction with a baseball, when you throw the ball, you exert a force on it. You exert an equal and opposite force on the earth when you throw the ball. If you are in space, and you throw the ball, the force you exert to throw the ball forward will also accelerate yourself backward.
Terminology for Momentum and Energy:
Momentum: P = m*v. Momentum is a vector of mass at a given velocity. Force = change in momentum over time. A high Force will change momentum quickly, and vice versa. Momentum has direction.
Work: W = F*s. Work is Force applied over a given distance. Work is energy. It has no direction.
Kinetic Energy: E = (1/2)*m*v^2. Kinetic energy is the energy of a body as it moves at a given velocity. When you accelerate a baseball to a certain velocity, you also impart kinetic energy to it.
Frictional Force: F = u*N. u (the greek letter mu) is the coefficient of friction. Friction is a force which acts 90 degrees from a force normal to a surface. It often converts kinetic energy into heat.
Law of Conservation of Momentum: The total momentum of a given system will always remain constant. The classic example of this is a bunch of billiard balls on a table. Let's say you, the cue, the balls, and the table, and the earth (ground) are one system. When you break the rack, you impart momentum to the ball with the cue (applying a force), and simultaneously apply force to the earth (newton's 3rd law), changing its momentum by that same amount. Then the cue ball hits the racked balls, it imparts momentum to first one, then other balls in a chain reaction. They impart momentum to each other but if you take the sum total of all of the momentum vectors of each ball, they will add up to the momentum of the single cue ball which you accelerated with the cue stick. At some point, frictional force between each ball and the felt/bumpers of the table will slow the balls down, and impart momentum back into the earth.
Law of Conservation of Energy: The total energy of a given system will always remain constant. Energy can never be destroyed--it can only change form. Again, take the billiard example. You impart kinetic energy into the cue ball via the cue stick, simultaneously imparting kinetic energy into the earth (newton's 3rd law). Energy has no vector, so you basically can think about each ball as having a number above it indicating how much kinetic energy is in each ball, and it goes down as the ball rolls on the felt, losing energy to friction. You also lose energy to internal friction within the bumpers, and to sonic energy for every "clack" you hear for an impact between balls. At some point, all the balls stop moving, with all kinetic energy lost to friction.
In this example energy started as chemical energy in your body, changed into kinetic energy in your muscles, which then transferred into the cue ball, and then all the other billiard balls. That energy then gradually gets lost to sonic energy (on impacts), and friction (on ball-to-ball impacts, to felt, and bumpers). Total energy of the system remains constant. net effect is your body loses chemical energy, and the environment gains heat (sound becomes heat after friction)
Potential Energy: Potential energy is the energy "stored" in an object. It is energy that once released, will often express itself as kinetic energy. One of the most common forms of this is gravitational potential energy. Another form is spring potential energy, where energy is stored in a spring. The tendons/ligaments in our bodies are springs, but while we may think of our bodies like coiled springs sometimes, this form of energy is actually not going to provide nearly as much energy as your muscles. Chemical energy can also be thought of as potential energy, as it is the energy of molecular bonds that is released so we can move and think.
Gravitational Potential Energy: E = m*g*h. This is the amount of gravitational energy stored in a body. Since Work = F*s, when you apply force to the body equal to or greater than gravity, in the opposite direction of gravity's pull, then the energy you add to the ball becomes stored as gravitational potential energy. For example, when you lift a baseball off the ground, you impart kinetic energy to the ball, until you stop moving it at, say, chest height. At that moment, the baseball has gravitational potential energy. When you let go of the ball, the potential energy becomes kinetic energy, due to gravity, and it drops down. After it bounces a few times and stops moving, all that energy is converted into heat, and dissipates into the environment.
Centripetal Acceleration: a = v^2 / r. This is the acceleration that is required to allow a point mass to travel in a circular trajectory. The direction of this acceleration is always towards the center of the circle. The velocity of the point mass is tangent to the circle at all times. In the stationary reference frame, if the centripetal acceleration is suddenly stopped (say because the string snapped), the point mass will continue moving in a straight line tangential to the circle, starting at the location of the point mass when the string broke. This is easily demonstrated using a sling, or any weight on a string. Spin the weight on the string, and let go. or have a friend do it, so you can see it clearly. it will launch away from the original circle in a tangent direction, starting when the string was let go.
Centrifugal Acceleration: This is a so-called virtual acceleration, that is the acceleration of a body within the rotating reference frame. In the rotating reference frame, this acceleration apparently "pushes" the body out of the circle. The true, centripetal acceleration is pulling inwards on the body to keep it rotating in a circle. Once the rotation stops, this virtual force disappears, and you fly along the tangent of the circle just like with centripetal acceleration.
My personal martial musings relating to these concepts
Taoist concepts of yin and yang, true and false, solid and insubstantial resonate strongly with newton's laws. Especially, for obvious reasons, the law of action-reaction, conservation of energy, and conservation of momentum.
The law of action-reaction resonates clearly with the section of the taiji treatise: 有上及有下. 有前則有後. 有左則有右. When there is up there is down, when there is front there is back, when there is left there is right.
Conservation of energy and momentum basically tell you that the power you generate in a punch, should be directly proportional with the amount your legs and body push off the earth, if you do it right.
The law of inertia encourages you to not get in the way of an opponent's momentum, but to step aside and guide it in the same direction they already started with. Also it encourages deflection of blows instead of directly getting in the way.
The concept of centripetal motion/acceleration is the same thing, in that if you grab someone's punch as they come to you, it is much easier to guide it in a circular arc just by "pulling inward", than it would be to knock it aside.
That should be about it for this chapter. Next chapter will be rotational/angular mechanics, torque, and conservation of angular momentum. Any questions are welcome. I will pose "test" questions to those of you who are interested.
Last edited by nianfong
on Wed Dec 14, 2011 2:08 pm, edited 20 times in total.