The relation between time t and displacement x is `t = alpha x^2 + beta x,` where `alpha and beta` are constants. The retardation is

Explanation

`t= alpha x^(2) + beta x`. Differntiating it `w.r.t.` time 9t) we have
`1 =alpha 2 x (dx)/(dt) + beta (dx)/(dt) =(dx)/(dt) (2 alpha x + beta)`
or ` (dx)/9dt) =v= 1/(2 alpha x + brta)`
Again differnting w.r.t. time (t), we have
` (dv)/(dt) = acceleration =(-2 alpha v)/((2 alpha x + beta)^(2) =- 2 alpha v^(2) ,`
so retadation ` =2 alpha v^(3)`.