Let’s see what happens when we start applying derivatives to those four fundamental concepts. Let’s start with position.

First let’s see how something’s position changes with time, ∂P/∂t. As stated above, we see that this is the object’s velocity. An important thing to note about velocity, however, is that it is a vector so it has two components, magnitude and direction. Thus, something can change its velocity by speeding up or slowing down, or by going at the same speed in a different direction.

Velocity [m/s] = Change in position over time. V=∂P/∂t

Now what happens if we take the derivative again? Acceleration is the term for how fast something’s velocity changes. Again, this can be a change in magnitude or direction. Acceleration is a key component in martial arts. If someone is fast, but fast all the time, then the opponent will be able to time them. If someone can change their speed or direction quickly, then they will be very difficult to beat.

Acceleration [m/s2] = Change in velocity over time. A=∂V/∂t

In order to find an object’s velocity, we can integrate the acceleration over time. We can integrate again (the velocity) to find an object’s position.

If we do the same thing with energy, we find that the change in energy with respect to time is called Power. But the change in energy is also referred to the as Work, so we get two definitions for Power.

Power [j/s or w] = Change in Energy over time. P=∂E/∂t

Power is measured in Watts where one Watt equals one Joule/second.

But what do we mean when we say that someone’s kick is powerful? We need a few more concepts to explain what happens when someone strikes another person. There are two critical things that happen- a transfer of momentum and a transfer and dissipation of energy.

The first important concept in collisions is called momentum. Momentum is what is transferred from one body to another when they run into each other. The other important thing to note is that Momentum is always conserved. This means that all of the Momentum transferred from one body is absorbed by the other body.

Momentum [kg m/s] = Mass times velocity m*v