एक प्रक्षेप्य को क्षैतिज से \(\theta\) कोण पर \(25\,m / s\) वेग के साथ प्रक्षेपित किया जाता है। \(t\) सेकण्ड पश्चात इसका क्षैतिज से झुकाव शून्य हो जाता है। यदि \(R\) प्रक्षेप्य की क्षैतिज परास है तो \(\theta\) का मान ज्ञात कीजिए-: \(\left[ g =10\,m / s ^2\right.\) लें]

A \(\frac{1}{2} \sin ^{-1}\left(\frac{5 t^{2}}{4 R}\right)\)
B \(\frac{1}{2} \sin ^{-1}\left(\frac{4 R }{5 t ^{2}}\right)\)
C \(\tan ^{-1}\left(\frac{4 t ^{2}}{5 R }\right)\)
D \(\cot ^{-1}\left(\frac{ R }{20 t ^{2}}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\cot ^{-1}\left(\frac{ R }{20 t ^{2}}\right)\)

Step-by-step Solution

Detailed explanation

\(R =\frac{ V ^{2}(2 \sin \theta \cos \theta)}{ g }\) \(t =\frac{ V \sin \theta}{ g } \Rightarrow V =\frac{ gt }{\sin \theta}\) \(\Rightarrow R =\frac{ g ^{2} t ^{2}}{\sin ^{2} \theta} \cdot \frac{2 \sin \theta \cos \theta}{ g }\)…

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

सूची-I का सूची-II से मिलान कीजिए।

सूची-I	सूची-II
(A) द्रव्यमान घनत्व	(I) [ML²T^-3]
(B) आवेग	(II) [MLT^-1]
(C) शक्ति	(III) [ML²T⁰]
(D) जड़त्व आघूर्ण	(IV) [ML^-3T⁰]

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