A particle is moving in sand under acceleration `a=g-lambday^(2)`, where lambda is a positive constant and `y` is the distance traveled on the y-axis. If the initial velocity is `v_(0)` and after traveling a distance `y_(m)`, the particle stops, find `lambda` (the motion is along the y-axis).

Explanation

`a=g-lambday^(2)`
`v(dv)/(dy)=g-lambday^(2)`
`int_(v_(0))^(0) v dv=int_(0)^(y_(m))(g-lambday^(2))dy`
`|v^(2)/2|_(v_(0))^(0)=|gy-(lambday^(3))/3|_(0)^(y_(m))`
`0-v_(0)^(2)/2=gy_(m)-(lambday_(m)^(3))/3`
`(lambday_(m)^(3))/3=gy_(m)+v_(0)^(2)/2=(2g y_(m)+v_(0)^(2))/2`
`lambda=(6 g y_(m)+3v_(0)^(2))/(2 y_(m)^(3))`