Question

Prove that Newton's second law of motion is the only real law of motion.

Answer 1

Step 1:

This can be demonstrated by the first and third laws of motion contained in the second law of motion.

Let us demonstrate that the first law of motion is contained in the second law of motion.

$\overrightarrow{F}=m\overrightarrow{a}$
$\therefore \overrightarrow{F}=0\Rightarrow \overrightarrow{a}=0$

That is, if there is no force acting on the body, its acceleration is zero. This means that if a body is at rest, it stays at rest, and if a body is moving in a straight line with constant velocity, it keeps moving. This is stated in the first law. As a result, the first law is contained in the second law.

Step 2:

Let us demonstrate that the Third Law is contained within the Second Law.

According to the second law of motion,

$\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$

i.e.,$\overrightarrow{F}=0$

$\Rightarrow \overrightarrow{p}$ = constant

Step 3:

Then, in the absence of external force, the total momentum of the system remains constant.
If we consider an isolated system of two bodies of masses $m_1$ and $m_2$. Let, vector ${\mathrm{F}}_{12}$ be the force exerted by mass ${\mathrm{m}}_1$ on ${\mathrm{m}}_2$ which is the action.
If, vector$F_{21}$ be the force exerted by mass $m_2$on $m_1$, which is the reaction.
If, due to ${\overrightarrow{F}}_{21}$,
Rate of change of momentum of $m_1=\frac{d{p}_1}{dt}$
Rate of change of momentum of ${\mathrm{m}}_2=\frac{{\mathrm{dp}}_2}{\mathrm{dt}}$
Then,

${\overrightarrow{F}}_{12}=\frac{d{p}_1}{dt}$and ${\overrightarrow{F}}_{21}=\frac{d{p}_2}{dt}$ Now, ${\overrightarrow{F}}_{12}+{\overrightarrow{F}}_{21}=\frac{d{p}_1}{dt}+\frac{d{p}_2}{dt}$

$=\frac{d}{dt}\left(p_1+p_2\right)$

$=\frac{d\overrightarrow{p}}{dt}$

Then, the vector $p$is the total momentum of the system.
According to the second law of motion, in the absence of external force, momentum remains conserved.
Therefore,
$\therefore {\overrightarrow{F}}_{12}+{\overrightarrow{F}}_{21}=\frac{d\overrightarrow{p}}{dt}=0$
$\Rightarrow {\overrightarrow{F}}_{12}=-{\overrightarrow{F}}_{21}$
It is the third law of motion that is proved.
Thus, Newton's second law of motion Is the real law of motion.