\(2 \times 10^4\,kg\) વજન ધરાવતું એક અંતરીક્ષયાન પૃથ્વીની સપાટીની નજીકની વર્તુળાકાર કક્ષામાં છોડવામાં આવે છે. ગુરુત્વાકર્ષીય ખેંચાણને દૂર કરવા માટે અવકાશયાનને કક્ષામાં આપવામાં આવતી વધારાનો વેગ ...... હશે. \(\left(\right.\) જો \(g =10\,m / s ^2\) અને પૃથ્વી ની ત્રિજ્યા \(\left.=6400\,km \right)\)

A \(11.2(\sqrt{2}-1)\,km / s\)
B \(7.9(\sqrt{2}-1)\,km / s\)
C \(8(\sqrt{2}-1)\,km / s\)
D \(7.4(\sqrt{2}-1)\,km / s\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(8(\sqrt{2}-1)\,km / s\)

Step-by-step Solution

Detailed explanation

\(v _{\text {orbit }}=\sqrt{\frac{ GM }{ R }}=\sqrt{ gR }\) \(v _{\text {escape }}=\sqrt{\frac{2 GM }{ R }}=\sqrt{2 gR }\) \(\Delta v =(\sqrt{2}-1) \sqrt{ gR }=8(\sqrt{2}-1)\,km / s\)

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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