4 ts.ithnkhehatofareatemeheastertoa. Show that v = √2gh, where g is the acceleration due togravity.b. By equating the rate of outflow to the rate of change of liquidin the tank, show that h(t) satisfies the equationdhA(h) at= -aa√√/2gh,(34)where A (h) is the area of the cross section of the tank at height hand a is the area of the outlet. The constant a is a contractioncoefficient that accounts for the observed fact that the crosssection of the (smooth) outflow stream is smaller than a. Thevalue of a for water is about 0.6.c. Consider a water tank in the form of a right circular cylinderthat is 3m high above the outlet. The radius of the tank is 1m,and the radius of the circular outlet is 0.1 m. If the tank is initiallyfull of water, determine how long it takes to drain the tank downto the level of the outlet. 4. Suppose that a tank containing a certain liquid has an outlet nearthe bottom. Let h(t) be the height of the liquid surface above the outletat time t. Torricelli's2 principle states that the outflow velocity v at theoutlet is equal to the velocity of a particle falling freely (with no drag)from the height h.2 Evangelista Torricelli (1608-1647), successor to Galileo as courtmathematician in Florence, published this result in 1644. In addition to thiswork in fluid dynamics, he is also known for constructing the first mercurybarometer and for making important contributions to geometry.

Answered: a. Show that v = √2gh, where g is the…

Computer Networking: A Top-Down Approach (7th Edition)

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Author:James Kurose, Keith Ross

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Related questions

Question

ts.
ith
nk
he
hat
of
are
ate
me
he
as
ter
to
a. Show that v = √2gh, where g is the acceleration due to
gravity.
b. By equating the rate of outflow to the rate of change of liquid
in the tank, show that h(t) satisfies the equation
dh
A(h) at
= -aa√√/2gh,
(34)
where A (h) is the area of the cross section of the tank at height h
and a is the area of the outlet. The constant a is a contraction
coefficient that accounts for the observed fact that the cross
section of the (smooth) outflow stream is smaller than a. The
value of a for water is about 0.6.
c. Consider a water tank in the form of a right circular cylinder
that is 3m high above the outlet. The radius of the tank is 1m,
and the radius of the circular outlet is 0.1 m. If the tank is initially
full of water, determine how long it takes to drain the tank down
to the level of the outlet.

4. Suppose that a tank containing a certain liquid has an outlet near
the bottom. Let h(t) be the height of the liquid surface above the outlet
at time t. Torricelli's2 principle states that the outflow velocity v at the
outlet is equal to the velocity of a particle falling freely (with no drag)
from the height h.
2 Evangelista Torricelli (1608-1647), successor to Galileo as court
mathematician in Florence, published this result in 1644. In addition to this
work in fluid dynamics, he is also known for constructing the first mercury
barometer and for making important contributions to geometry.

Expert Solution

Step 1

(a) The fundamental kinematic equation of motion is used to create the equation v = √(2gh), where v is the speed at which an object is falling to the ground, h is the height from which it is falling, and g is the acceleration caused by gravity.

When an object is dropped from a height h, gravity will cause it to fall freely, and the equation states that the velocity v will grow linearly with time t:

v = gt

The distance traveled during this time is given by the following equation:

h = gt^2 / 2

Rearranging the equation and solving for t, we get:

t = √(2h/g)

Substituting this value of t back into the first equation, we get:

v = gt = g √(2h/g) = √(2gh)

Since g is the acceleration brought on by gravity and h is the height from which an object is falling, the final expression for the velocity at which it is falling to the ground is v = √(2gh).