Question

prove that f is equals to M a

Answer 1

Let , a body has mass , m..... an initial velocity , u .....and a force , f acts on the body , in a time , t and it causes a final velocity , v..
since, force is directly proportional to change in momentum / time taken .

therefore, f = mv - mu / t
f = m ( v- u ) / t
f = ma

Answer 2

let an object is having mass = m
let it's velocity changes to 'u' to 'v' in the time interval 't'

means here,

initial velocity = u
final velocity = v
we know that,
momentum (p) = mass × velocity
So,
momentum (p1) of the object at it's initial velocity = m ×u = mu
momentum (p2) of the object at it's final velocity = m × v = mv

change in momentum = (p1) - (p2) = mv - mu
the rate of change of momentum =
(mv - mu)÷t ---------------(1)

According to newton's second law of motion force is directly proportional to the rate of change of momentum.

after substituting the rate of change of momentum from equation (1) we get :-
force is directly proportional to (mv - mu)÷ t
force is directly proportional to m (v - u ) ÷ t
force is directly proportional to m×a
here ( a = (v - u) ÷ t)

F = k ma ----------------(2)

here, k is proportionality constant.
since, 1 unit force is defined as the mass of 1 kg object produces acceleration of 1m/s^2.
therefore, 1 unit force = k × 1kg × 1m/s^2
thus k = 1
by substituting the value of 'k = 1' in equation (2) we get :-

F = ma
Hence, proved.