Question

derive Newton's first law of motion from mathematical expression of the symbol of motion

Answer 1

Newton's First Law of Motion:

The Newton's first law of motion states that an object continues to be in the state of rest or in the state of uniform motion along a straight line, until an unbalance force is not acting upon the object.

Now, let's talk about Newton's Second Law of Motion

Newton's second law states that the rate of change of momentum is directly proportional to the fore applied to change the object's momentum. Mathematically, it can be written as $F = \frac{m(v-u)}t = ma \\ or, \: F = ma$

Newtons First Law is a special case of Newton's second law!!!

According to Newton's Second Law of motion, $F = ma$.

If, F = 0 N, Then,

$0 = \frac{m(v-u)}t \\ \implies m(v-u) = 0 \\ \implies v-u = 0 \\ \implies v = u \\ or, \: u = v$

This is what the first law says. If an object is in the state of uniform motion along a straight line, then it will continue to do so until an unbalanced force acts upon it.

Further,

If, u = 0, then v = 0 --because u = v, If F = 0.....

This means that an object will continue to be in its state of rest until an external unbalanced force acts upon it.

Hence, Newton's first law can be derived from his second law of motion.......