Question

Prove f = ma

Answer!

Answer 1

Newton's 2 law :- Rate of Change of momentum is directly proportional to Force applied. 

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Given rate of change of momentum is directly proportional to its Force applied. 

First of all

Momentum =P = mv 

Rate of change of velocity = Pf - Pi /t

Here Pi is the initial momentum, 

Pf is the final momentum 

now

Pf - Pi /t ∝F 

mv- mu /t ∝ F 

we know that final and initial velocities are considered v,u respectively. 

m( v - u)/t ∝F 

F ∝ ma 

here we take acceleration = v-u/t

So for removing proportionality I am introducing a proportionality constant Say K 

finally F= maK 

here K =1 

Finally F = m* a 

Thus, after deriving, it gives definition of force. 
Force is the product of the mass of the object and acceleration produced. 

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hope helped!

Answer 2

Newton’s 2nd Law:
Force acting on a body moving is directly proportional to its rate of change of momentum.

Mathematically,
F ∝ dp / dt
F = kdp / dt
F = dp / dt ……[∵ It was found that proportionality constant k is equal to 1]
F = d(mv) / dt ……[1]

If only velocity is changing per time then equation [1] will become
F = m dv/dt
F = ma ……[∵ dv/dt = a]
E.g. It is apllicable for a body when it’s pushed by a force “F”

If only mass is changing per time then equation [1] will become
F = v dm/dt
E.g. It is applicable in case of conveyer belt where velocity is constant.