Question

Write mathematical formulation of Newton's second law of motion

Answer 1

AnSwer :

$\textsc{Mathematical\:formulation\:of\:Newton's\: second\:law\:of\:motion\::-}$

◑ Suppose,

• m = mass of a body
• u = initial velocity of the body along a straight line
• F = an external force applied on the body
• t = time for which the force is applied
• v = final velocity of the body along the same straight line, after t stand

Initial linear momentum of the body, $\sf{p_1 = mu}$

Final linear momentum of the body, $\sf{p_2 = mv}$

Change in linear momentum of the body, $\sf{p_2 - p_1 = mv - mv = m(v - u) }$

Rate of change of linear momentum = change in linear momentum/time taken

$\sf{\dfrac{m(v - u)}{t}}$

But, $\sf{\dfrac{(v - u)}{t}}$ = $\sf{\dfrac{change\:in\:velocity}{time\:taken}}$ = rate of change of velocity = acceleration of the body = a

Therefore, rate of change of linear momentum = ma

◑ According to Newton's second law of motion, rate of change of linear momentum $\sf{\propto}$ force applied.

that is, ma $\sf{\propto}$ F or F $\sf{\propto}$ ma or F $\bf{\propto}$ kma (1)

where k is a constant of proportionality. The value of k depends on the units adopted for measuring the force.

◑ If we define F = 1, when m = 1 and a = 1, i.e., unit force is that much force which produces unit acceleration in s body of a unit mess, then from required equation :-

→ 1 = k × 1 × 1

→ k = 1

◑ Putting in (1), we get :-

$\boxed{\sf F = ma}$

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Answer 2

Answer:

F=ma is the mathematical formulation of Newtown's second law of motion