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Review. Old Faithful Geyser in Yellowstone National Park erupts at approximately one-hour intervals, and the height of the water column reaches 40.0 m (Fig. P15.40). (a) Model the rising stream as a series of separate droplets. Analyze the free-fall motion of one of the droplets to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoulli’s equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 m? Assume the chamber is large compared with the geyser’s vent.
Figure P15.40
(a)
The speed at which the water leaves the ground by modelling the stream as a series of separate droplets.
Answer to Problem 40P
The speed at which the water leaves the ground by modelling the stream as a series of separate droplets is
Explanation of Solution
Consider the upward flight of a water-drop projectile from geyser vent to fountain top.
Take the upward direction to be
Write the equation of motion in the vertical direction.
Here,
At the maximum height the velocity of the water drop will be zero so that the value of
Substitute
Conclusion:
The value of
Substitute
Therefore, the speed at which the water leaves the ground by modelling the stream as a series of separate droplets is
(b)
The speed of the water as it leaves the ground by modelling the stream as an ideal fluid in streamline flow.
Answer to Problem 40P
The speed of the water as it leaves the ground by modelling the stream as an ideal fluid in streamline flow is
Explanation of Solution
Write the Bernoulli’s equation.
Here,
Air has very low density so that the pressure at both geyser vent and fountain-top will be atmospheric pressure.
Substitute
Conclusion:
Substitute
Therefore, the speed of the water as it leaves the ground by modelling the stream as an ideal fluid in streamline flow is
(c)
The comparison of answers of part (a) and part (b).
Answer to Problem 40P
The answers of part (a) and part (b) agree precisely.
Explanation of Solution
It is asked to find the speed with which the water leaves the ground. In part (a), the rising stream was modelled as free-fall motion of one of the droplets. The value of the speed found using this model is
In part (b), the rising stream was modelled as an ideal fluid in streamline flow. The value of the speed found using this model is also
Conclusion:
Thus, the answers of part (a) and part (b) agree precisely.
(c)
The pressure in the heated underground chamber.
Answer to Problem 40P
The pressure in the heated underground chamber is
Explanation of Solution
Take point 1 in the Bernoulli’s equation to be the chamber and the point 2 to be fountain-top.
The velocity of the water drop at both the points are
Substitute
Conclusion:
The density of water is
Substitute
Therefore, the pressure in the heated underground chamber is
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