# physic mechanic

A particle is pushed horizontally by a constant horizontal force of magnitude F, which
starts from the rest at x=0 in the positive ݔ direction. During the movement, it is also
acted by the air resistance force that equivalent to bv, where b is the positive coefficient
and v is the instantaneous velocity of the body.

Find out the acceleration of object at any time t.

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1. F - bv = m a

a = (F-bv)/m

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2. that means

dv/dt = F/m - (b/m)v
or
(b/m) v + dv/dt = F/m

let v = c (1-e^-kt)
then dv/dt = cke^-kt
and
(b/m)c -(b/m)ce^-kt +cke^-kt = F/m
when t --> oo, F = bc so c = F/b
when t = 0, c k = F/m
so Fk/b =F/m
so k = b/m
now
v = (F/b)( 1-e^-(bt/m) )
a = dv/dt = (F/b)(b/m)e^-(bt/m)
or
a = (F/m)e^-(bt/m)

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