Water stand at a heigh H in a tank whose sides are vertical. An orifice is made at the depth h on one of the wall Fig. `7(c).25`. The emerging stream of water strikes at the distance R from the tank on the floor./ Find the relation for R in term of h, and H. When R is maxmimum, how h and H are related.

Answer

[`2sqrt(h(H-h)); h=H//2`]

Explanation

The velocity with which the liquid flows out of the orifice is `v=sqrt(2gh)` or `v^(2)=2 gh` The liquid flowing out of the orifice, becomes a projected horizontally. Height of orifice from ground `=(H-h)`. If is the the time taken by the water to reach the ground, then
`t=sqrt(2(H-h)l g),`
Horizontal range,
`R=vt=sqrt(2gh)xxsqrt((2(H-h))/(g))=2sqrt(h(H-h))`
For R to be maximum, `(dR)/(dh)=0`
`:.(dR)/(dh)=2xx(1)/(2)[h(H-h)]^((-1)/(2)) (d)/(dh)(H h-h^(2))`
`=(1)/(sqrt(h(H-h)))9H-2h=0`
`:. H-2 h=0` or `h=H//2`