Mass and Collisions

For self-defense, mass is huge. In fact, mass should be one of the primary things anyone should think about when he or she sizes up a potential opponent. The issue here is that a smaller person will have a disadvantage going after a larger person. The problem is not speed or strength per se, but mass. A larger person will more than likely be stronger than a smaller person, and added to this advantage is that a larger person will have more mass and more momentum.

As I showed previously, the power (i.e. force) a person generated is a combination of mass and speed. Your foot or fist has a certain effective mass (the mass of the striking point plus the amount of weight behind the attack) and a velocity. Look at the following example:

There are two people, one that can bring 20kg to bear for a punch, and the other guy that can bring 10kg to the table. To generate the same amount of momentum, the smaller person will have to be moving twice as fast as the larger person. This seems reasonable. In order to get something moving, a martial artist needs to be able to exert force on their muscles in order to accelerate them. A smaller person will have less mass to move, and so he requires less force. If they reach 30m/s in 0.4 seconds, then the acceleration of the punch is:

We can then estimate the force required for each punch.

This means the small guy only needs to generate half as much force to get his fist to 30m/s in 0.4 seconds than the larger guy. This is good for the smaller guy, because it means that he may be faster, since less of his force is required to accelerate the punch. Speed is essential for any smaller opponent, as it is the only thing that gives the smaller person an advantage when both opponents are equally skilled.

So where does the larger person's advantage come from? It's all that extra mass. To show why, the term "elastic collision" is one of those phrases that means a specific type of physical event, and explicitly a collision of two "particles" of a given mass and velocity, respectively. Just imagine two spheres heading straight toward each other until they hit and bounce off. Their final velocity will be determined by their initial velocity and their mass. For our purposes, we'll use a specific equation that assumes the velocity of the object getting hit starts out at 0 (i.e. it's not moving.) Though this won't be a perfect demonstration, the idea is that this is emulating a strike.

Equation 6:

Equation 6 calculates the end velocity of the stationary particle (m2) when it is hit by the moving particle (m2). Let's say that m1 = 100kg, m2 = 200kg, and vi1 = 20 m/s.

The larger particle would be moving 13.3 m/s after the collision. Let's reverse it, and see how the smaller particle fairs when it is hit by the larger particle.

Interestingly, the smaller particle is moving faster than the larger particle after the collision (The larger particle will be moving at 6.6m/s. Given hugely different masses, the smaller particle would at most travel twice as fast as the initial velocity of the larger particle, which would hardly be affected).

For a martial artist, this means that in order to generate the same amount of power a smaller person will have to move twice as a fast as a larger person. It is definitely true that a smaller person can move more quickly than a larger person, but what's the real difference? It's not easy to graph speed versus body mass, because the speed of a person is dependent on several factors, such as physical fitness, stockiness, muscle mass, and how densely packed the muscles are. In general, it follows a line closer to a logarithmic curve. From the example above, I showed that a particle with half the mass had to travel twice as fast to generate the same power. The thing is that a 50kg person will not be twice as fast as a 100kg person, given that both are of equal physical fitness. This is difficult to quantize accurately, but essentially the ability for a smaller person to generate an equal amount of power is directly proportional to how fast he or she can move.

In theory, a smaller person could break the same number of boards (or bones, I guess) as a larger person given that the correct amount of speed was achieved. However, there are other factors the work against the little guy. Weight and sheer strength play a huge part, and a larger person will have the upper hand in this case. While not a sure thing, its still a safe bet to assume that should things get nasty, a larger person will be more likely to be able to life the smaller person than vice versa. If it comes down to strength vs. strength, then again the larger person has an advantage. This doesn't mean things are hopeless for the a smaller combatant, but it does mean there are things they need to be aware of. This is one of the reasons there are weight classes in professional fighting. The larger opponent wouldn't always win, but it certainly be more likely.