A particle having charge 's' starts from rest in a region where the electric field has constant direction but magnitude varying with time `t` as `E=E_(0)t` .In time `t`, it is observed that the particle acquire a velocity `v` after covering a distance `x` then

Options

A. `x=(1)/(2)(E_(0)s)t^(3)`
B. `x=(1)/(6)(E_(0)s)t^(3)`
C. `v=(E_(0)s)t^(3)`
D. `v=(1)/(2)(E_(0)s)t^(2)`

Answer

B,D

Explanation

`E=E_(0)t` since `a=(qE)/(m)=(qE_(0)t)/(m)=E_(0)st`
`rArr (dv)/(dt)=E_(0)st rArr dv=E_(0)stdt`
`rArr v=E_(0)s int_(0)^(t)tdt=(1)/(2)E_(0)st^(2)rArr (dx)/(dt)=(1)/(2)E_(0)st^(2)`
`rArr dx=(1)/(2)E_(0)st^(2)dt`
`rArr int_(0)^(x)dx=(1)/(2)E_(0)s int_(0)^(t)t^(2)dt rArr x=(1)/(6)E_(0)st^(3)`