# How to derive 1st of motion from second law of motion?

Feb 26, 2018

So,Newton's $2 n d$ Law of motion states that, force $F$ acting is equals to the rate of change in momentum $\frac{\delta P}{\delta t}$

So, if a particle of mass $m$ was initially moving with velocity $u$ changed its velocity to $v$ in time $t$,its change in momentum ($\delta P$)= $m \left(v - u\right)$

So, $\frac{\delta P}{\delta t} = \frac{m \left(v - u\right)}{t - 0} = \frac{m \left(v - u\right)}{t}$

So, $F = \frac{m \left(v - u\right)}{t}$

Now, if $F = 0$,

$\frac{m \left(v - u\right)}{t} = 0$

so, $v - u = 0$

so, $v = u$

and if $u = 0 , v = 0$

That means,when no force is acting, a body that was initially at rest i.e $u = 0$,will remain at rest,so $v = 0$

Or,if it was moving with constant velocity $u$,it will be moving with the same velocity i.e $v = u$(means, final velocity ($v$) = initial velocity($u$)

This is the Newton's $1$ st Law of motion.

Feb 26, 2018

You can't!

#### Explanation:

While substituting $F = 0$ in $F = m a$ does give $a = 0$, this is by no means a proof of Newton's first law from the second. Note that a basic law in physics is, at its heart, an admission of defeat - it is a scientist's way of saying that these (the basic laws) can not be proven - they have to be taken on trust - but once you accept them you can prove the rest of the results by logical deduction. In this sense they are akin to axions in geometry.

That Newton, who was one of the smartest human beings ever to have lived, passed up this chance to cut down his basic laws from three to two - should be an indication that this "proof" is not quite right!

Actually Newton's second law is valid only for a special class of frames called inertial frames. And a frame is inertial only if Newton's first law is valid in it. So, before you apply second law, you have to already make sure that the first law is valid. So, the argument that is often given (wrongly) of the proof of the first law from the second is circular - you can not prove the first law by assuming that it holds to begin with!