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

Options

A. `2 alpha v^(3)`
B. `2 beta v^(3)`
C. `2 alpha beta v^(3)`
D. `2 beta^(2) v^(3)`

Answer

A

Explanation

`t = alpha x^(2) + beta x`
Differentiating w.r.t. 't'
`1 = alpha . 2x (dx)/(dt) + beta (dx)/(dt)`
`rArr (dx)/(dt) = v = ((1)/(2 alpha x + beta))`....(i)
Again differentiating w.r.t. 't'
`0 = alpha .2 ((dx)/(dt))^(2) + 2 alpha x (d^(2))/(dt^(2)) + beta (d^(2)x)/(d t^(2))`
`.: (dx)/(dt) = v`
`:. (d^(2)x)/(dt^(2)) = (-2 alpha v^(2))/((2 alpha x + beta)) = - 2 alpha v^(3)`