A particle moves in the `xy` plane with a constant acceleration `omega` directed along the negative y-axis. The equation of motion of particle has the form `y = cx -dx^(2)`, where `c` and `d` are positive constants. Find the velocity of the particle at the origin of coordinates.

Explanation

Proceed as in the previous problem, here `omega = g`
`= sqrt((omega(1 + c^(2)))/(2d)), theta = tan^(-1) ( c)`
OR
`y = c x - d x^(2)`
`(d y)/(d t) = c(d x)/(d t) - d.2x(d x)/(d t)`
`v_(y) = c v_(x) - 2 d x v_(x)`
`(d v_(y))/(d t) = c(d v_(x))/(d t) - 2 d [x(d v_(x))/(d t) + v_(x).(d x)/(d t)]`
`a_(y) = c.a_(x) - 2 d[x a_(x) + v_(x)^(2)]`
`a_(x) = 0, a_(y) = - omega`
`- omega = - 2d v_(x)^(2) rArr v_(x) = sqrt((omega)/(2 d))` (indepedent of `x`)
`v_(y) = c v_(x) - 2d x v_(x)`
At origin,
`x = 0, v_(y) = c v_(x) = csqrt((omega)/(2 d))`
`v = sqrt(v_(x)^(2) + v_(y)^(2))`
`= sqrt((omega)/(2 d) + (c^(2) omega)/(2 d)) = sqrt((omega(1 + c^(2)))/(2 d)`
`tan theta = (v_(y))/(v_(x)) = c rArr = theta ta^(-1)( c)`