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Newton's Laws

We are now ready to state Newton's three "Laws" of motion, in Newton's own words:

1.
FIRST LAW: Every body continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by a force impressed on it.
2.
SECOND LAW: The change in motion [rate of change of momentum with time] is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
3.
THIRD LAW: To every action there is always opposed an equal reaction; or, the mutual actions of two bodies are always equal, and directed to contrary parts.
Now, Newton's language was fairly precise, but to our modern ears it sounds a bit stilted and not very concise. We also imagine that, with the benefit of several centuries of practice, we have achieved a clearer understanding of these Laws than Newton himself. Regardless of the validity of this conceit, we like to express the Laws in a more modern form with a little mathematical notation thrown in:
1.
FIRST LAW: A body's velocity [which might be zero] will never change unless and until a force acts on the body.
2.
SECOND LAW: The time rate of chage of the momentum of a body is equal to the force acting on the body. That is,

 (5)

3.
THIRD LAW: Whenever a force is applied to A by B, there is an equal and opposite reaction force on B due to A. That is,

 (6)

where the subscript AB (for instance) indicates the force from A to B.

As long as the mass m is constant7 we have

since the derivative of a constant times a variable is the constant times the derivative of the variable. Then the SECOND LAW takes the more familiar form,

 (7)

This famous equation is often written in scalar form,

because and are always in the same direction.

Next: What Force? Up: Force vs. Mass Previous: Momentum
Jess H. Brewer - Last modified: Fri Nov 13 17:32:47 PST 2015