For a particle moving along a circular path the radial acceleration `a_r ` is proportional to `t^2` (square of time) . If `a_z` is tangential acceleration which of the following is independent of time

Options

A. `a_t`
B. `a_r.a_t`
C. `a_r/a_t`
D. `a_r(a_t)^2`

Answer

D

Explanation

`a_(r)=v^(2)/R= kt rArr v=sqrt(Rkt) rArr _(t)=(dv)/(dt)=1/2 sqrt((Rk)/(t))`
`rArr a_(t)^(2)=1/4 (Rk)/(4) rArr t=(Rk)/(4a_(t)^(2)) and a_(r)=k (Rk)/(4a_(t)^(2) rArr a_(r)a_(t)^(2)=(kRk)/(4)`