Question

explain Newton's second law mathematically explaining the symbols used

Answer 1

F∝ mv-mu/t     {Force is directly proportional to change in momentum and indirectly proportional to time}

F = (k) (mv - mu) / t    { k = constant of proportionality}

F = (k) (m) (v-u)/t

F = (k) (m) (a)   { (v-u)/t = acceleration = a}

F = ma {Experimentally, k=1}

Hope this helps!

Answer 2

$\textbf{Explanation of Newton's Second Law Mathematically}$


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,

$\textbf{F = ma}$
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