Consider a rod of length l resting on a wall and the floor. If the velocity of the upper end of a rod A is `v_(0)` in downward direction, find the velocity of the lower end B at this iinstant, when the rod makes an angle `theta ` with horizontal

Explanation

`x^(2)+y^(2)=l^(2)`
Differentiating w.r.t. time t, we get
`2x(dx)/(dt)+2y(dy)/(dt)=2l(dl)/(dt)`
`(dl)/(dt)=0` (since the length l of the rod is constant)
`x(dx)/(dt)+y(dy)/(dt)=0`
`(dx)/(dt)=v_(B),(dy)/(dt)=-v_(0)`
[As the increases, x increases, y decreases]
`(dx)/(dt)+ve,(dy)/(dt)=-ve`
`xv_(B)-yv_(0)=0impliesv_(B)=(y)/(x)v_(0)`
`v_(B)=v_(0)tantheta`
OR
As the rod is rigid the velocity of point A and B along the length of the rod will be same.
`v_(B)=costheta=v_(0)sintheta`
`v_(B)=v_(0)tantheta`