A More Advanced Look at Newton's Laws

A more formal way to look at Newton's laws for the advanced student is as follows:

First Law: A body remains at rest or in uniform motion unless acted upon by a net force.

This helps to define the basic force law, f=ma, where f is the net force acting on the object. You can have many different forces acting on an object, but if they all cancel each other out, then there is no net force and therefore no acceleration. The equation f=ma defines a force as a mass times an acceleration. Note that acceleration can be a change in either speed or direction. Given any two of force, mass, and acceleration, the third can be found. Because acceleration is the first derivative of velocity and the second derivative of position, this equation can be rewritten in terms of those quantities.

Second Law: A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

Written as an equation, the second law becomes:


where p is the momentum and t is time.

This is another way of stating f=ma. Because p=mv and mass does not change, it follows directly. This gives a much more quantifiable method of calculating momentum conservation.

Third Law: If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

This law, for all its complex repercussions, is as simple as it sounds. There are some specific instances when it does not apply, however. One example of this, although too complex to discuss here, is the force between moving point charges.