Two planets P1 and P2 with equal mass have radii R1 and R2, respectively, where R2=R1/2. The escape speeds of P1 and P2 are v1 and v2, respectively. Then v2/v1 is :

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Physics Q13 – NEET-UG 2026

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Home Physics GravitationQuestion #201

Two planets

P_{1}

and

P_{2}

with equal mass have radii

R_{1}

and

R_{2}

, respectively, where

R_{2} = \frac{R_{1}}{2}

. The escape speeds of

P_{1}

and

P_{2}

are

v_{1}

and

v_{2}

, respectively. Then

\frac{v_{2}}{v_{1}}

is :

Clue 1 of 3: Escape Velocity Concept

Escape velocity is the minimum speed required for an object to escape the gravitational influence of a massive body without further propulsion.

🥷

Ninja Strategy•Proportionality and Magnitude Check

Recognize that escape velocity is inversely proportional to the square root of the radius. A smaller radius ( $R_{2}$ ) for the same mass implies a larger escape velocity ( $v_{2}$ ), thus $\frac{v_{2}}{v_{1}}$ must be greater than 1. This eliminates options (1) and (2). The square root dependence helps distinguish between $\sqrt{2}$ and $2$ .

🧠 Full Solution Path

Step 1: Recall the formula for escape velocity✦ Active

The escape velocity $v_{e}$ from a planet of mass $M$ and radius $R$ is given by:

v_{e} = \sqrt{\frac{2 G M}{R}}

Step 2: Apply the formula to both planets and form the ratio○ Expand

Given that both planets have equal mass $M$ , the escape speeds for $P_{1}$ and $P_{2}$ are:

v_{1} = \sqrt{\frac{2 G M}{R_{1}}}

v_{2} = \sqrt{\frac{2 G M}{R_{2}}}

Now, form the ratio $\frac{v_{2}}{v_{1}}$ :

\frac{v_{2}}{v_{1}} = \frac{\sqrt{\frac{2 G M}{R_{2}}}}{\sqrt{\frac{2 G M}{R_{1}}}} = \sqrt{\frac{R_{1}}{R_{2}}}

💡 Teacher's Secret Hint

Notice that $2 G M$ cancels out, simplifying the ratio significantly.

Step 3: Substitute the given relationship between radii○ Expand

We are given that $R_{2} = \frac{R_{1}}{2}$ . Substitute this into the ratio:

\frac{v_{2}}{v_{1}} = \sqrt{\frac{R_{1}}{R_{1} / 2}} = \sqrt{2}

💡 Teacher's Secret Hint

A smaller radius for the same mass leads to a higher escape velocity.

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Physics Question 13 – NEET-UG 2026

Clue 1 of 3: Escape Velocity Concept

🧠 Full Solution Path