Question

write the mathematical form of Newton's second law of motion .

Answer 1

$\huge\mathfrak{Mathematical\:form\:of}$
$\huge\mathfrak{Newton's\:Second\:Law}$

We have to find mathematical form of Newton's Second Law means F = ma.

Force = (Change in momentum / Time)

Now, we know that;

Momentum = Product of mass and velocity of the body. i.e.

P = mv

Let us consider that a body of mass (m) is moving with a initial velocity (u) and with final velocity (v) in time (t)

Initial momentum = p1 = mu

Final momentum = p2 = mv

Then,

Change in momentum of the body = p2 - p1

F = mv -mu

F = [m (v - u) / t] .........(1)

As we know that,

a = [(v - u) / t] ...........(2)

Put the value of (2) in (1)

Then,

$\textbf{F = ma}$
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OR

Rate of change of linear momentum is directly proportional to force.

So,

F α dp/dt 

F = k dp/dt

Now,

p (momentum) = mv

So,

F = k d(mv)/dt

F = k md(v)/dt

F = k ma

(As, dv/dt = a)

Experimentally k =1

So,

$\textbf{F = ma}$

$<marquee>$$<b><u>Hence, proved...$
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Answer 2

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