Question

State Newton's second law of motion and show that the first law of motion can be mathematically stated from the Newton's seconds law.

Answer 1

Newton's first law of motion:

• It states that “Every object continues to be in the state of rest or uniform motion in same direction unless a net external force is acted upon it.”

Newton's second law of motion:

• It states that “The rate of change of momentum of a body is directly proportional to the force applied, and takes place in the same direction in which the force acts.”
• According to this law, $Force\propto Rateofchangeinmomentum$.

Deriving first law of motion from second law of motion:

• According to son law, Momentum is the product mass and velocity, $p=mv$.
• So, the rate of change of momentum can be written as $∆p=mv-mu$, where v is the final velocity and u is the initial velocity.
• Then, rate of change of momentum will be

$\begin{array}{rcl}Rateofchangeofmomentum& =& \frac{∆p}{t}\\ & =& \frac{mv-mu}{t}\\ & =& m\frac{v-u}{t}\end{array}$

• This can be further written as $\begin{array}{rcl}Rateofchangeofmomentum& =& ma\end{array}$, since $\frac{v-u}{t}=a$, where $a$ is the acceleration.
• Thus it is obtained that, “Net force applied on a body is equal to rate of change of momentum”, $F=\begin{array}{rcl}& & \frac{∆p}{t}\end{array}$, since force is the product of mass and acceleration, $F=ma$.
• This implies, if net force is zero, the body will continue to move with same velocity.
• This leads to the first law of motion, according to which until external force is applied, the body of mass m moves with uniform velocity.

Hence, Newton's first law of motion can be mathematically stated from the Newton's seconds law.