Question

derive the equation F = m a​

Answer 1

Explanation:

From Newton's second law of motion

Rate of change of momentum is directly proportional to applied unbalanced force and take place in the same direction in which force is applied.

Therefore,

F ∝ Rate of change of momentum

F ∝ (mv-mu)/t

F∝ [m (v-u)]/t                (taking m common)

F ∝ m × (v-u)/t ...........................(i)

Since the rate of change in velocity is known as acceleration

Therefore, a= (v-u)/t .....................(ii)

Substituting the value of (ii) in (i)

We get, F ∝ m × a

When we remove ∝ this sign we put 1 as constant.

So, F = 1 × m × a

F= m × a

 

Answer 2

Explanation:

According to newton's second law of motion ,

' the rate of change of momentum of a body is directly proportional to the applied force '

Mathematically , the force , F is proportional to delta p divided by t

where delta p= change in momentum and

t = time interval

let an object of mass 'm' is moving along a straight line with an initial velocity 'u' . It is accelerated uniformly to velocity 'v' in time 't' by the application of constant force F throughout the time t.

the initial and final momentum of an pbject will be

$p_1 = mu$

AND

$p_2 = mv$

The change in momentum ,delta p= p2 - p1

mv - mu

= m(v-u)

As change of momentum is proportional to force F

then F is proportional to m(v-u)/t

$f \: = k \times \frac{m(v - u)}{t} ...........(1)$

(where k = constant of proportionality)

Now , acceleration ,

$a \: = \frac{( \: v - u)}{t} ............(2)$

From eq.(1) and eq.(2) , we get

F = k m a .............(3)

The SI units of mass and acceleration are kg and ms^-2 respectively. The unit of force us so choosen that the value of 'k' becomes one.

That is , 1 unit of force = k × (1 kg) × ( 1 ms^-2)

or k = 1

Thus the value of k becomes 1.

therefore , the eq.(3) reduces to ,

F = ma

I know it is little bit long answer but I hope this helps you!!!

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