# Can you derive newton's 1st and 3rd laws from f=ma, and the conservation of energy.

1. May 19, 2012

### aabaa

The first law seems obvious, plug in 0 for acceleration and you get 0 force. Therefore for something to accelerate there needs to be some net force.

Also, wouldn't the third law have to be true if energy is conserved at all times?

Sorry if this has been asked before.

2. May 21, 2012

### Cleonis

Re: Can you derive newton's 1st and 3rd laws from f=ma, and the conservation of energ

Generally, a lot of laws are interderivable.

However, it's far from elegant to try and derive from energy conservation. Let me illustrate why by comparing perfectly elastic collision with inelastic collision. In the case of elastic collision both momentum and kinetic energy are conserved. But in the case of inelastic collision some of the kinetic energy transforms to internal energy (usually heat). Momentum is always conserved, kinetic energy not always. In that sense momentum is the more general element.

I take it you feel that Newton's laws of motion are not optimal; I take it you are looking for a way to state the same content with less elements.

Here is a way of stating the same things in a form with two laws, rather than three.

1 .The principle of uniformness of space.
For position and velocity: space has the same properties everywhere and in all directions. Position-, velocity- and acceleration vectors add according to vector addition in euclidean space.

2. The relation between exerted force and change of velocity
F=ma
At every point in space, in every direction, change of velocity relative to space is proportional to exerted force.

I take it that it's clear that the above two principles imply Newton's first and second law. It may not be as straightforward to see that the third law is implied as well.

Let me try and clarify with an example: a tug of war contest.
First the case where both teams can dig their heels in the ground. Digging in their heels is what gives each team the leverage to exert a pulling force on the other team.
Now imagine that both teams are on a surface that is perfectly frictionless. Do the teams still have leverage when there is zero friction? They do! There's always inertia. If team A has twice as much mass as team B, and team A pulls in the rope, then both teams move, but team B moves twice as much.
That is: how much team A will move and how much team B will move is described by F=ma

So now we have the same content as Newton's laws, but this time in two principles.

Point of criticism:
One may argue that the principle of uniformness of space on one hand, and the rules of addition of position, velocity and acceleration on the other hand, are actually two distinct principles.

In fact, in special relativity (introduced in 1905) they are distinct principles. Special relativity asserts uniformness of space just as in newtonian mechanics, but the rules for addition of position-, velocity- and acceleration vectors are different.

3. May 21, 2012

### Whovian

Re: Can you derive newton's 1st and 3rd laws from f=ma, and the conservation of energ

One thing. The First Law (Newton's, not Thermodynamics) is easily derivable from Newton's Second Law. But the Third Law is effectively a restatement of Conservation of Momentum, and has nothing to do with Conservation of Energy.