A particle starts from the origin of coordinates at time t = 0 and moves in the xy plane with a constant acceleration `alpha` in the y-direction. Its equation of motion is `y= betax^2`. Its velocity component in the x-direction is

Options

A. `sqrt((alpha)/(beta))`
B. `sqrt((beta)/(alpha))`
C. `sqrt((alpha)/(2 beta))`
D. `sqrt((beta)/(2 alpha))`

Answer

D

Explanation

`y = alpha x^(2) rArr (d y)/(d t) = alpha.2x (d x)/(d t)`
`v_(y) = 2 alpha x v_(x)`
`(dv_(y))/(d t) = 2 alpha(x(d v_(x))/(d t) + v_(x).(d x)/(d t))`
`a_(y) = 2 alpha(x a_(x) + v_(x)^(2))`
`a_(x) = 0, a_(y) = beta`
`beta = 2 alpha v_(x)^(2) rArr v_(x) = sqrt((beta)/(2 alpha))`
OR
This equation is similar to the equation of trajectory of a projectile thrown horizontally from the top of the tower.
`y = alpha x^(2), y = (g x^(2))/(2 u^(2)) = (beta x^(2))/(2 u^(2))`
`alpha = (beta)/(2 u^(2)) rArr u = sqrt((beta)/(2 alpha))`