સમય \(t\) સાથે કણનું સ્થાન \(x\left( t \right) = at+ b{t^2} - c{t^3}\) મુજબ આપવામાં આવે છે, જ્યાં \(a, b\) અને \(c\) અચળાંક છે જ્યારે કણનો પ્રવેગ શૂન્ય થાય ત્યારે તેનો વેગ કેટલો હશે?

A \(a + \frac{{{b^2}}}{{4c}}\)
B \(a + \frac{{{b^2}}}{{c}}\)
C \(a + \frac{{{b^2}}}{{2c}}\)
D \(a + \frac{{{b^2}}}{{3c}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(a + \frac{{{b^2}}}{{3c}}\)

Step-by-step Solution

Detailed explanation

\(\begin{array}{l} X = at + b{t^2} - c{t^3}\\ V = \frac{{dx}}{{dt}} = a + 2bt - 3c{t^2}\\ a = \frac{{dv}}{{dt}} = 2b - 6ct\\ Put\,acceleration\, = 0\\ \Rightarrow \,\,t = \frac{b}{{3c}}\\ Find\,V\,at\,t = \frac{b}{{3c}}\\ V = \,a + \frac{{{b^2}}}{{3c}} \end{array}\)

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.

Same subject

સૂચિ - \(I\) અને સૂચિ - \(II\) મેળવો.

	સૂચિ - \(I\)		સૂચિ- \(II\)
\(A\).	સ્નિગ્ધતા અંક	\(I\).	\([M L^2T^{–2}]\)
\(B\).	પૃષ્ઠ તાણ	\(II\).	\([M L^2T^{–1}]\)
\(C\).	કોણીય વેગમાન	\(III\).	\([M L^{-1}T^{–1}]\)
\(D\).	ચાક ગતિ ઉર્જા	\(IV\).	\([M L^0T^{–2}]\)

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