પૃથ્વીની સપાટીથી (પૃથ્વીની ત્રિજ્યા \(6.4 \times 10^3 \,km\)) \(h\) ઊંચાઈ પર એક ઉપગ્રહને રાખવા માટેની જરૂરી ઊર્જા \(E_1\) છે અને આ ઉપગ્રહને આ ઊંચાઈ પર વર્તુળાકાર કક્ષામાં રાખવા જરૂરી ગતિ ઊર્જા \(E_2\) છે. \(E_1\) અને \(E_2\) સમાન થાય તેવી ઊંચાઈ \(h\) નું મૂલ્ય છે

A \(1.6\times 10^3\,km\)
B \(3.2\times 10^3\,km\)
C \(6.4\times 10^3\,km\)
D \(1.28\times 10^4\,km\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(3.2\times 10^3\,km\)

Step-by-step Solution

Detailed explanation

\({E_1} =-\frac{{GMm}}{{R + h}} - \left( { - \frac{{GMm}}{R}} \right)\) \({E_2} = \frac{1}{2}m{\left( {\sqrt {\frac{{GM}}{{R + h}}} } \right)^2} = \frac{{GMm}}{{2\left( {R + h} \right)}}\) \({E_1} = {E_2}\,\,;\,\,h = \frac{R}{2}\)

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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