Question

# A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by ( left(4 t^{3}-2 tright), ) where ( t ) is in second and velocity is in ( mathrm{m} / mathrm{s} ). What is acceleration of the particle, when it is at distance ( 2 mathrm{m} ) from the origin.

Solution

Velocity of the particle at any instant ( t, v=4 t^{3}- ) ( 2 t )

=> Acceleration at any instant t

( a=frac{d v}{d t}=12 t^{2}-2 )

Position of the particle at any instant t

( mathrm{dx}=left(4 mathrm{t}^{3}-2 mathrm{t}right) mathrm{dt} )

( int d x=intleft(4 t^{3}-2 tright) d t )

( =>x=t^{4}-t^{2} )

( mathrm{At} mathrm{x}=2 mathrm{m} )

( t^{4}-t^{2}-2=0 )

( =>left(t^{2}+1right)left(t^{2}-2right)=0 )

t cannot be imaginary, therefore, ( t^{2}=2 )

Substituting this value of ( t^{2} ) in equation (1)

( =>mathrm{a}=22 mathrm{m} / mathrm{s}^{2} )