Statement I: For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. [2023]

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Q.
Given below are two statements: [2023]

Statement I: For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases.

Statement II: Escape velocity is independent of the radius of the planet.

In the light of above statements, choose the most appropriate answer from the options given below:

1 Both Statement I and Statement II are incorrect

2 Statement I is correct but Statement II is incorrect

3 Statement I is incorrect but Statement II is correct

4 Both Statement I and Statement II are correct

Ans.
(2)

$V_{e} = \sqrt{\frac{2 G M}{R}} \Rightarrow V_{e} \propto \sqrt{\frac{M}{R}}$

$As \frac{M}{R} increases \Rightarrow V_{e} increases$

$Statement (1) is correct$

$Also, V_{e} \propto \frac{1}{\sqrt{R}}$

$As V_{e} depends upon R$

$\Rightarrow Statement (2) is incorrect$

Solution

1 Both Statement I and Statement II are incorrect

2 Statement I is correct but Statement II is incorrect

3 Statement I is incorrect but Statement II is correct

4 Both Statement I and Statement II are correct

Ans.
(2)

$V_{e} = \sqrt{\frac{2 G M}{R}} \Rightarrow V_{e} \propto \sqrt{\frac{M}{R}}$

$As \frac{M}{R} increases \Rightarrow V_{e} increases$

$Statement (1) is correct$

$Also, V_{e} \propto \frac{1}{\sqrt{R}}$

$As V_{e} depends upon R$

$\Rightarrow Statement (2) is incorrect$

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Solution

1 Both Statement I and Statement II are incorrect

2 Statement I is correct but Statement II is incorrect

3 Statement I is incorrect but Statement II is correct

4 Both Statement I and Statement II are correct

Ans. (2) Ve=2GMR⇒Ve∝MR As MR increases ⇒Ve increases Statement (1) is correct Also, Ve∝1R As Ve depends upon R ⇒ Statement (2) is incorrect

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Ans.
(2)

$V_{e} = \sqrt{\frac{2 G M}{R}} \Rightarrow V_{e} \propto \sqrt{\frac{M}{R}}$

$As \frac{M}{R} increases \Rightarrow V_{e} increases$

$Statement (1) is correct$

$Also, V_{e} \propto \frac{1}{\sqrt{R}}$

$As V_{e} depends upon R$

$\Rightarrow Statement (2) is incorrect$