\(m\) द्रव्यमान का कोई पिण्ड, \(\lambda v _{ e }\) वेग से धरातल से आसमान में ऊर्ध्वाधर ऊपर की ओर प्रक्षेपित किया जाता है। \(v _{ e }\) पलायन वेग है, एवं \(\lambda < 1\) है। यदि हवा का प्रतिरोध नगण्य है, तो यह पिण्ड पृथ्वी के केन्द्र से कितनी अधिकतम ऊँचाई तक जाएगा? (\(R:\) पृथ्वी की त्रिज्या)

A \(\frac{R}{1+\lambda^{2}}\)
B \(\frac{R}{1-\lambda^{2}}\)
C \(\frac{R}{1-\lambda}\)
D \(\frac{\lambda^{2} R}{1-\lambda^{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{R}{1-\lambda^{2}}\)

Step-by-step Solution

Detailed explanation

\(-\frac{ GMm }{ R }+\frac{1}{2} m \lambda^{2} V _{ e }^{2}=-\frac{ GMm }{ h }\) \(-\frac{ GMm }{ R }+\frac{1}{2} \lambda^{2} \frac{2 GMm }{ R }=-\frac{ GMm }{ h }\) \(\frac{\lambda^{2}}{ R }-\frac{1}{ R }=\frac{-1}{ h }\) \(\frac{1}{ h }=\frac{1-\lambda^{2}}{ R }\)…

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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