Question

Which of the following is the most significant law of motion given by Newton?
A) First Law of motion
B) Second Law of motion
C) Third law of motion
D) Zeroth law of motion

Answer 1

It is known to us that there are three laws of motion formulated by Sir Isaac Newton. Here, in the question, the most significant law implies which law can define the other two laws when some conditions are applied to them. So, here we have to derive the other two laws of motion using only one law of motion.

Complete step by step answer:
Newton’s second law of motion states that the rate of change of momentum of a body is directly proportional to the force applied on it and the momentum occurs in the direction of the net applied force. Mathematically, it can be represented as:
$F \propto \dfrac{{dp}}{{dt}}......(1)$
Where,
$F$ is the force.
$p$ is the momentum.
By recollecting our knowledge of momentum, we get that, $p = mv$ where $m$ is the mass and $v$ is the velocity of the object. Substituting in equation (1) we get,
$F = \dfrac{{d(mv)}}{{dt}}$
$\Rightarrow F = m\dfrac{{dv}}{{dt}}$
$\Rightarrow F = ma......(2)$
Where,
$a$ is the acceleration.
We will now use the equation of motion to get the value of acceleration:
$v = u + at$
Where,
$v$ is the final velocity.
$u$ is the initial velocity.
$a$ is the acceleration.
$t$ is the time.
$\Rightarrow v - u = at$
$\Rightarrow a = \dfrac{{v - u}}{t}......(3)$
Substituting equation (3) in (2) we get,
$F = m\left( {\dfrac{{v - u}}{t}} \right)$
$\Rightarrow Ft = mv - mu......(4)$

Now, let us consider the force applied to the object to be zero. So, equation (4) becomes,
$0 = mv - mu$
$\Rightarrow mv = mu$
$\Rightarrow v = u$
This shows that the object continues to move with the same velocity in the absence of force.
Also, when the object is initially at rest, i.e. $u = 0$ and $F = 0$ , then the final velocity is also zero, i.e.$v = 0$. This shows that an object at rest continues to be at rest if no force is acting on it.
Thus, the first law of motion is derived using the second law of motion.
Let us consider an isolated system having two bodies $A$ and $B$. Let ${F_{AB}}$ be the force acting on $B$ due to $A$ and ${F_{BA}}$ be the force acting on $A$ due to $B$.
Also, rate of change of momentum of $A$ is $\dfrac{{d{p_A}}}{{dt}}$ and rate of change of momentum of $B$ is $\dfrac{{d{p_B}}}{{dt}}$. We will use the second law of motion to get,
${F_{AB}} = \dfrac{{d{p_B}}}{{dt}}.......(5)$
${F_{BA}} = \dfrac{{d{p_A}}}{{dt}}.......(6)$
We will now add equations (4) and equation (5) to get,
${F_{AB}} + {F_{BA}} = \dfrac{{d{p_B}}}{{dt}} + \dfrac{{d{p_A}}}{{dt}}$
$\Rightarrow {F_{AB}} + {F_{BA}} = \dfrac{d}{{dt}}\left( {{p_A} + {p_B}} \right)......(7)$
But, we have considered an isolated system where no external force is applied to the object. Therefore, momentum will be zero as there is no velocity. So,
$\dfrac{d}{{dt}}\left( {{p_A} + {p_B}} \right) = 0$
$\Rightarrow {F_{AB}} + {F_{BA}} = 0$
Equation (8) gives the third law of motion using the second law.
Therefore, option (B) is the correct answer as it can describe the other two laws.

Note:
Newton’s first law of motion states that if a body is in the state of rest or in motion with a constant speed in a straight line, then the body will remain in the state of rest or in motion unless and until an external force is applied on it. But, in our daily life, we observe that an object in motion eventually comes to rest or an object at rest will start to move. It is due to the presence of different external forces present in our environment. For example, if we kick a ball, it moves a certain distance and comes to rest due to an external force called friction. In space, when an object is in motion, it continues to be in motion as there are no external forces present there.