Question

In a cylindrical water tank here are two small holes Q and P on the wall at a depth of $h_1$ from the upper level of water and at a height of $h_2$ from the lower end of the tank, respectively, as shown in the figure. Water coming out from both the holes strike the ground at the same point. The ratio of $h_1$ and $h_2$ is

• A. $1$
• B. $2$
• C. $>1$
• D. $<1$

Answer 1

The correct option is A $1$

The two streams strike at the same point on the ground.
$R_1=R_2=R$
$u_1{t}_1=u_2{t}_2$
where $u_1$ velocity of efflux at $Q=\sqrt{\left(2g{h}_1\right)}$ and $u_2=$ velocity of efflux at $P=\sqrt{\left[2g\right(H-h_2\left)\right]}$
$t_1=$ time of fall of water stream through Q is
$\sqrt{\frac{2(H-h_1)}{g}}$
$t_2=$ time of fall of the water stream through $P=\sqrt{\frac{2h_2}{g}}$
Putting theses values is Eq. (i), we get
$\sqrt{2g{h}_1}\left(\sqrt{\frac{2(H-h_1)}{g}}\right)=\sqrt{2g(H-h_2)}\sqrt{\frac{2h_2}{g}}$
$(H-h_1)h_1=(H-h_2)h_2$
or $[H-(h_1+h_2\left)\right][h_1-h_2]=0$
$H=h_1+h_2$ is irrelevant because the holes are at two different heights. Therefore, $h_1=h_2$ or $h_1/h_2=1$