આકૃતિમાં દર્શાવ્યા પ્રમાણે પદાર્થ અનુક્રમે \(AB, BC\) અને \(CD\) રેખાખંડ પર \(v_1, v_2\) અને \(v_3\) ઝડપથી ગતિ કરે છે. જ્યાં \(AB = BC\) અને \(AD =3 AB\), તો પદાર્થની સરેરાશ ઝડપ કેટલી હશે?

A \(\frac{\left(v_1+v_2+v_3\right)}{3}\)
B \(\frac{v_1 v_2 v_3}{3\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}\)
C \(\frac{3 v_1 v_2 v_3}{v_1 v_2+v_2 v_3+v_3 v_1}\)
D \(\frac{\left(v_1+v_2+v_3\right)}{3 v_1 v_2 v_3}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{3 v_1 v_2 v_3}{v_1 v_2+v_2 v_3+v_3 v_1}\)

Step-by-step Solution

Detailed explanation

\(AB =x\) \(BC =x\) \(2 x+ CD =3 x\) \(CD =x\) \(\ < v > =\frac{3 x}{\frac{x}{v_1}+\frac{x}{v_2}+\frac{x}{v_3}}=\frac{3 v_1 v_2 v_3}{v_2 v_3+v_1 v_3+v_1 v_2}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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