પૃથ્વીના નિષ્ક્રમણ વેગ અને ગ્રહના નિષ્ક્રમણ વેગનો ગુણોત્તર કેટલો થાય? આપેલ : ગ્રહનું દળ પૃથ્વીના દળ કરતાં \(16\) ગણું અને ગ્રહની ત્રિજ્યા પૃથ્વીની ત્રિજ્યા કરતાં \(4\) ગણી છે.

A \(4:1\)
B \(2:1\)
C \(1: \sqrt{2}\)
D \(1: 4\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(2:1\)

Step-by-step Solution

Detailed explanation

\(V_{\text {escape }}=\sqrt{\frac{2 G M}{R}}\) \(\therefore V_{\text {escape }} \text { for planet }=\sqrt{\frac{2 G\left(16 M_E\right)}{\left(4 R_E\right)}}=2 \sqrt{\frac{2 G_E}{R_E}}\) \(=2\left(V_{\text {escape }} \text { for Earth }\right)\)

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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