\(H\) ઊંચાઈ પરથી મુકત પતન કરતો એક પદાર્થ, \(h\) ઊંચાઈ પર આવેલા એક ઢોળાવ વાળા સમતલ સાથે સંપૂર્ણ સ્થિતિસ્થાપક રીતે અથડાય છે. અથડામણ બાદ પદાર્થનો વેગ સમક્ષિતિજ થાય છે. જો આ પદાર્થ જમીન પર પહોંચવા માટે મહત્તમ સમય લેતો હોય તો \(\frac{\mathrm{H}}{\mathrm{h}}\) નું મૂલ્ય _______ છે.

A \(2\)
B \(3\)
C \(4\)
D \(5\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(2\)

Step-by-step Solution

Detailed explanation

Total time of flight \(=\mathrm{T}\) \(T=\sqrt{\frac{2 h}{g}}+\sqrt{\frac{2(H-h)}{g}}\) For \(\max\) time \(=\frac{d T}{d h}=0\) \(\sqrt{\frac{2}{g}}\left(\frac{-1}{2 \sqrt{\mathrm{H}-\mathrm{h}}}+\frac{1}{2 \sqrt{\mathrm{h}}}\right)=0\)…

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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