JEE Physics 2023 Question with Solution

Given: Two statements about escape velocity.

Find: Which statement is correct.

The formula for escape velocity is

$$v_e = \sqrt{\frac{2GM}{R}}$$

where $G$ is the gravitational constant, $M$ is the mass of the planet, and $R$ is the radius of the planet.

From this formula,

$$v_e \propto \sqrt{\frac{M}{R}}$$

So, as the ratio $\frac{M}{R}$ increases, the escape velocity $v_e$ increases. Hence, Statement I is correct.

Also, $v_e$ depends on $R$ as seen from the formula, so escape velocity is not independent of the radius of the planet. Hence, Statement II is incorrect.

The solution states that the correct answer is $\boxed{(3)}$ and marks Option C, but this conflicts with the derived conclusion above. Based on the working, the defensible correct option should be B.

Given:

• Escape velocity formula
• Statement I concerns increase of $\frac{M}{R}$
• Statement II claims independence from $R$

Find: Truth values of the two statements.

Using

$$v_e = \sqrt{\frac{2GM}{R}}$$

we note that escape velocity increases when $M$ increases for fixed $R$, and decreases when $R$ increases for fixed $M$.

Therefore, the quantity controlling the trend is

$$\frac{M}{R}$$

inside the square root. If $\frac{M}{R}$ increases, then $v_e$ also increases. So Statement I is true.

Because $R$ appears explicitly in the denominator, escape velocity does depend on radius. So Statement II is false.

Therefore, the correct option by the physics working is B.