Question 198: Properties of Matter

Question

Consider a water jar of radius R that has water filled up to height H and is kept on astand of height h (see figure). Through a hole of radius r (r << R) at its bottom, the water leaks out and the stream of water
coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is x. Then :

Accepted Answer

v
1
= velocity of water when it leak from hole
v
2
= velocity of water when it reach the ground.
From Bernoulli's principle,
$${1 \over 2}ho {v_1}^2 + ho gh$$ = $${1 \over 2}ho {v_2}^2$$
$$ \Rightarrow $$   $${v_1}^2$$ + 2gh = $${v_2}^2$$
From Torricelli's theorem,
v
1
= $$\sqrt {2gH} $$
$$ 	herefore $$   $${v_2}^2$$ = 2gh + 22gH
From continuity equation,
$${A_1}{v_1}$$ = $${A_2}{v_2}$$
$$ \Rightarrow $$   $$\pi {r^2} 	imes \sqrt {2gH} $$ =  $$\pi {x^2}\sqrt {2g\left( {h + H} ight)} $$
$$ \Rightarrow $$   x
2
=  r
2
$$\sqrt {{H \over {H + g}}} $$
$$ \Rightarrow $$   x = r $${\left( {{H \over {H + g}}} ight)^{{1 \over 4}}}$$

Answer

$$x = r\left( {{H \over {H + h}}} \right)$$

Answer

$$x = r{\left( {{H \over {H + h}}} \right)^{{1 \over 2}}}$$

Answer

$$x = r{\left( {{H \over {H + h}}} \right)^{{1 \over 4}}}$$

Answer

$$x = r{\left( {{H \over {H + h}}} \right)^{{2}}}$$

Properties of Matter

Choose the correct answer: