In the previous chapter we figured out that forces cause change in motion. In this chapter we will try to quantify this relationship – put it into an equation. The first thing we need to be aware of in order to do so, is that while forces cause acceleration (change in motion), there is something that resists this change in motion.
In football there are certain players who by design the coach does not want to move. They are supposed to protect the quarterback by staying planted wherever they are. Can you guess what is so special about these players? Let me tell you that they do not have to be especially handsome  they are in fact especially heavy. It’s the big players, the ones that look like refrigerators that do this job. Their big mass makes sure that they will not move from the place they are put in. Their mass resists their change in motion – since they are standing still, they will most likely continue to stay still and not topple over. The same is true once they get going. You better get out of the way if a big guy like that come barreling down toward you, nothing will stop them from keeping going. Their mass resists the change in motion. While forces cause change in motion (or acceleration) mass resists the change in motion. 
We just figured out that the three aspects that we have to investigate are forces, mass and acceleration. We will use the simple example of a sled to figure out the relationship between these three variables. Assume you push a light sled with as much force as you can muster and then do the same with a heavy sled. It will be obvious that the light sled will speed up easier, will have more acceleration. This makes sense because mass resists acceleration, the more mass the less acceleration (if the force is the same). The two are inversely proportional: . The sign in the middle is the sign indicating proportionality. The indicates the inverse character of the relationship – more mass means less acceleration. If you increase the mass “m” in the denominator the whole fraction gets smaller (think of ). Now take the same sled with the same mass and push it once just a little and then again with all your might. In the second example the sled will accelerate more. Force and acceleration are directly proportional or . Keep in mind that we are talking about acceleration here. Pushing with the same force means that you have to keep running faster and faster. In reality this would not be possible. Most of the time you are only pushing the sled up to a certain speed and then maintain that speed. So we have and . If we bring those two expressions together we get .

This does still not mean that both sides are equal  they are proportional. In order to make them equal we have to multiply in a proportionality factor. Here is an example of what this means. If you are going in a car at a constant speed both time and distance are proportional. The more time passes the more distance you will have covered. But they are not equal. Numerically. If you will have gone for one hour you won’t have covered only one mile, or after three hours, three miles. Distance and time will always be proportional but in order to make them equal you need a proportionality factor. If you are going 40 mph this factor is 40. After one hour you will have gone 40 miles, after two hours 80 miles, etc..
A factor like this is needed in our force, acceleration, mass problem as well. This factor was rather complicated, but the use of the metric system made things much simpler. Now the factor is simply “1”. This is, however, not by default but by design. Force, mass and acceleration and their units are designed so that the factor turns out to be “1”. The unit of force is Newton, and 1 N is defined as the force necessary to accelerate 1 kg to 1 m/s/s. All of this makes things much simpler and turns to or . The latter is the form in which the relationship between forces, mass and acceleration is most often written. Matchbox are models that are built in proportion. Everything on a Matchbox car is made in proportion, the wheels, the chassis, even the seats. 