Question

Show that Newton’s second law is the real law of motion.

Answer 1

We are asked to verify the statement that Newton's second law is the real law of motion. For this, we have to make sure that we are able to formulate all the other laws of motion only from the second law of motion. In other words, we have to deduce all the formulation of all the fundamental laws of motion taking the initiating formulation as Newton's second law.

Formulae Used:
$F = ma$
Where, $m$ is the mass of the body, $a$ is the acceleration of the body and $F$ is the unbalanced external force applied on the body.
$F = \dfrac{{dp}}{{dt}}$
Where,$p$ is the linear momentum of the body.

Complete solution:
Firstly, We will discuss the first law of motion.
As per definition, the first law of motion states that a body at rest remains to be at rest or a body in uniform linear motion remains to stay in uniform linear motion until and unless an external force acts on it. In other words, a body tends to retain its motion type until and unless an unbalanced external force acts on it.
Now, We will use the newton’s second law as
$F = ma$
Now, Let $F = 0$.
Then, Either, $m = 0$
Or, $a = 0$
But for an actual body, $m \ne 0$
Thus, $a = 0$
Thus, we get
If $F = 0$, then, $a = 0$ which means if there is no unbalanced external force acting on an isolated body, then, the body tends to retain its motion.
Now, We will discuss the law of conservation of momentum.
This states that the total linear momentum remains constant for an isolated body.
Now, We will use the newton’s second law as
$F = \dfrac{{dp}}{{dt}}$
Now, For an isolated body,$F = 0$
Thus, we get $\dfrac{{dp}}{{dt}} = 0$
For this to be true, $p$ has to be constant.
Hence, $p = constant$
Thus, we come up with the law of conservation of momentum.
Then, we will discuss the third law of motion which states that every action has equal and opposite reactions acting on different surfaces.
Now, We will use the newton’s second law as
$F = ma$
Now, Let $a = - a$
Substituting this, we get $F = m\left( { - a} \right)$
Further, we get $F = - ma$
Then, we get $F = - F$
Thus, giving us Newton's third law of motion.
Hence, as we can deduce all the fundamental laws of motion just from slight value changes in the formulation of the newton’s second law of motion, we can say this as the real law of motion.

Note: For the derivation of the first law of motion we inferred the law from $a = 0$. This is because if a body retains its motion, then the velocity of the body remains constant and thus makes acceleration equal to zero.