Question

state newtons second law of motion and show the first law of motion can be mathematically stated from the mathematical expression for the second law of motion .

Answer 1

Dear Student,

Newton's 2nd Law of Motion states that the rate of change of momentum of a body is always equal to the net force applied to the body.
Mathematically, $$\begin{gathered} F_{net}=\frac{∆mv}{t}=\frac{m∆v}{t}=ma \\ {F}_{net}=ma \end{gathered}$$

Newton's 1st Law states that If a Body is in a state of rest or non-accelerated motion, it will stay in it until and unless an external force is applied.
Mathematically,
When Body is in a state of rest or non-accelerated motion, a= 0
Thus,$$\begin{gathered} F_{net}=ma=m\left(0\right) \\ {F}_{net}=0 \end{gathered}$$
Which means there is no external force to change the motion of the body.

Or when External force of magnitude "F" is applied to the body.
Then, $$\begin{gathered} F_{net}=ma \\ a=\frac{F}{m} \end{gathered}$$
The body will change its motion with acceleration "a" in effect of net external force applied to it.
Hence, Newton's 1st Law of motion is proved mathematically.

Regards

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