KCET · Physics · Gravitation

A space station is at a height equal to the radius of the Earth. If ' \( V_{E} \) ' is the escape velocity on the surface of the Earth, the same on the space station is __ times \( V_{E} \).

A \( \frac{1}{2} \)
B \( \frac{1}{4} \)
C \( \frac{1}{\sqrt{2}} \)
D \( \frac{1}{\sqrt{3}} \)

Verified Solution

Answer & Solution

Correct Answer

(C) \( \frac{1}{\sqrt{2}} \)

Step-by-step Solution

Detailed explanation

Given, a space station is at a height equal to radius of Earth, \( \frac{1}{2} M v_{s}^{2}=\frac{G M m}{(R+h)} \)
Here, \( h=R \) and \( v_{s} \) is velocity of space station
\(\Rightarrow \frac{1}{2} M v_{s}^{2}=\frac{G M m}{2 R} \)
\(\Rightarrow v_{s}=\sqrt{\frac{G M}{R}}\)
Now, we know that escape velocity is
\(v_{e}=\sqrt{\frac{2 G M}{R}}=\sqrt{2} \sqrt{\frac{G M}{2}} \)
\(\Rightarrow v_{e}=\sqrt{2} v_{s} \)
\(\text { or } v_{s}=\frac{1}{\sqrt{2}} v_{e}\)

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

List-I	List-II
(a)	Angle of reflection	(i)	\(60^\circ\)
(b)	Value of \(\alpha\)	(ii)	\(120^\circ\)
(c)	Angle of deviation	(iii)	\(30^\circ\)

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