Deriving Formulas for Vmin & Vmax on Glare Ice w/ Zero Friction

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SUMMARY

The discussion focuses on deriving formulas for minimum speed (Vmin) and maximum speed (Vmax) for a car negotiating a circular curve on glare ice, where the coefficient of static friction (µ) is zero. The key equations involve the radius of the curve (R), gravitational acceleration (g), and the speed of the car (v). The user attempted to derive Vmin using the formula Vsqrt((1-mu(Rg)/(V^2))/(1+mu(V^2)/(Rg))), but received feedback indicating the solution was incorrect. Accurate derivation of Vmin and Vmax is essential for understanding vehicle dynamics on slippery surfaces.

PREREQUISITES
  • Understanding of circular motion dynamics
  • Knowledge of friction coefficients in physics
  • Familiarity with basic algebra and equation manipulation
  • Concept of gravitational force and its effects on motion
NEXT STEPS
  • Derive the correct formulas for Vmin and Vmax using the principles of circular motion.
  • Study the effects of varying the coefficient of static friction (µ) on vehicle stability.
  • Explore the role of gravitational acceleration (g) in determining safe speeds on curves.
  • Review examples of vehicle dynamics on slippery surfaces for practical applications.
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and vehicle dynamics, as well as engineers involved in road design and safety assessments.

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Homework Statement



A circular curve of radius R in a new highway is designed so that a car traveling at speed v can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmin to vmax. Derive formulas for vmin and vmax using mu for µ, and v, R, and g as appropriate.

Vmin= ?
Vmax= ?

Homework Equations


How can i derive a formula for this?


The Attempt at a Solution


I've tried a couple but I don't know exactly how to do it. When i put my answer it comes out wrong. help?
 
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I got for vmin:
Vsqrt((1-mu(Rg)/(V^2))/(1+mu(V^2)/(Rg)))
but it keeps saying its wrong!??!?
 

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