Show that v 2 is the smaller root of the quadratic equation v 2 2...

Ismael Molina

Ismael Molina

Answered

2022-07-24

Show that v 2 is the smaller root of the quadratic equation
v 2 2 2 σ M v 1 l 1 v 2 [ v 1 2 2 σ M l 1 ] = 0

Answer & Explanation

gardapati5u

gardapati5u

Expert

2022-07-25Added 9 answers

v 2 2 2 σ M v 1 l 1 v 2 [ v 1 2 2 σ M l 1 ] = 0
By using roots method
v 2 = ( 2 σ M v 1 l 1 ) ± ( 2 σ M v 1 l 1 ) 2 4 [ 1 ] [ v 1 2 + 2 σ M l 1 ] 2 1 v 2 = + 2 σ M v 1 l 1 ± ( 2 σ M v 1 l 1 ) 2 + 4 v 1 2 4 2 σ M l 1 2 v 2 = 2 σ M v 1 l 1 ± ( 2 v 1 2 σ M v 1 l 1 ) 2 v 2 = 2 σ M v 1 l 1 + 2 v 1 2 σ M l 1 v 1 v 2 = 2 v 1 v 2 = 2 σ M v 1 l 1 + 2 v 1 2 σ M l 1 v 1 v 2 = 2 v 1 v 2 = 2 σ M V 1 l 1 2 V 1 + 2 σ M v 1 l 1 v 2 = 2 2 σ m v 1 l 1 2 v 1
Therefore for the second roots v 2
as we increases the value of v 1
the v 2 will decreases due to two factor
1. negative stop " 2 v 1 "
2. exponential decreasing function " α 1 v 1 "Therefore v 2 < v 1 proved.

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