Can a Waterfall Ever Become Supersonic?

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SUMMARY

The discussion centers on calculating the minimum height required for a waterfall to achieve supersonic speeds, specifically exceeding 340 m/s, the speed of sound. The key equations utilized include the kinematic equation S = ut + 1/2 at² and the velocity relationship v = v₀ + gt. The solution involves solving a quadratic equation derived from these relationships, emphasizing the need to express time in terms of height to find the necessary height for supersonic flow.

PREREQUISITES
  • Understanding of kinematic equations, specifically S = ut + 1/2 at²
  • Knowledge of gravitational acceleration (g = 9.8 m/s²)
  • Familiarity with the concept of supersonic speed and the speed of sound (340 m/s)
  • Basic algebra skills for solving quadratic equations
NEXT STEPS
  • Study the derivation and application of kinematic equations in physics
  • Learn about the principles of fluid dynamics and how they relate to waterfall mechanics
  • Explore the effects of air resistance on falling objects and its impact on speed calculations
  • Investigate real-world examples of waterfalls and their flow rates to understand practical applications
USEFUL FOR

Students in physics or engineering, educators teaching kinematics, and anyone interested in fluid dynamics and the physics of motion.

spacethisride
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Homework Statement


According to geologists, the water in a waterfall was supersonic, that is, it fell with speeds in excess of the speed of sound. Ignoring air resistance, what is the minimum height necessary to create a supersonic waterfall? (The speed of sound may be taken to be 340 m/s.)

Homework Equations


Treating water as particles, water falling down the fall is acted upon by gravity. Using the Newton's equation...I think??

S = ut + 1/2 at^2

The Attempt at a Solution



S = ut + 1/2 at^2

h = 340t + (9.8/2)t^2
4.9t^2 +340t - h = 0. Solve the quadratic equation for t?

But I have two variables...(h and t) so I don't know how to go any further...
 
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you need to use the veloctiy relationship

[tex]v = v_0 + gt[/tex]

to find the time required to reach 340 m/s then use that time in the distance relationship to find the distance.
 

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