Escape Speed NEET 2026: Half-Radius Ratio Explained

What is the correct answer to the escape speed NEET 2026 question?

Option C is correct in the escape speed NEET 2026 question: halving a planet’s radius at fixed mass increases its surface escape speed by the square root of two.

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

The source is NEET 2026, Code-50, Physics, Gravitation. Planet P₁ has radius R₁. Equally massive planet P₂ has half that radius. Their surface escape speeds are v₁ and v₂. Find the second speed divided by the first.

The question bank classifies this as medium, with a 45-second target, not measured student performance. No figure is needed: these are independent planets with no spatial arrangement to interpret.

How do you write the escape-speed formula for both planets?

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

The symbols under the square root denote the gravitational constant, planet’s mass and planet’s radius, respectively. This formula gives escape speed at the surface in the usual isolated-planet model, neglecting atmospheric resistance and rotation. Use the same mass in both expressions because the planets are equally massive.

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

The official solution uses the inverse-square-root dependence at fixed mass. Escape speed is not inversely proportional to radius itself:

v \propto \frac{1}{\sqrt{R}}

Form the requested ratio, second speed divided by first. Dividing by the first expression puts its radius in the numerator:

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

The gravitational constant, the factor two and the common mass cancel. No numerical planetary data are required. Keep the radius ratio inside the square root when substituting.

How do you substitute the radius and check the answer?

Insert the half-radius condition only after forming the ratio. Keep the first planet’s radius on top and replace the second planet’s radius in the denominator. This keeps the calculation tied to the requested speed ratio rather than its reciprocal.

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

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

The answer is option C, not option D. For the same mass, the smaller planet has the greater surface escape speed, so the requested ratio must exceed one.

This direction check rejects A and B but cannot distinguish C from D. The square root is still essential.

Check units too: a speed divided by a speed is dimensionless. The square root of the radius ratio is also dimensionless, so the result passes that check.

Why does option D come from a method error?

Option D follows from dropping the square root, not from reversing the ratio. The radius change is handled correctly, but the dependence of speed on radius is changed. Here is the incorrect chain:

R_{2} = \frac{R_{1}}{2} ⟹ \frac{v_{2}}{v_{1}} = \frac{R_{1}}{R_{2}} = 2 (incorrect)

The precise error is replacing the inverse-square-root dependence with an inverse-radius dependence. These are different laws:

v \propto R^{- 1 / 2} replaced incorrectly by v \propto R^{- 1}

At fixed mass, the square of escape speed, not escape speed itself, is inversely proportional to radius. Halving the radius doubles the squared speed, so repair the calculation as follows:

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

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

Take the positive square root because speeds are non-negative. Write the exponent before substituting the radius change. This preserves the distinction between doubling a speed and doubling its square.

How do you solve three related escape-speed ratio questions?

Use the fixed-mass shortcut only when the masses are equal. Otherwise, keep both mass and radius ratios. These are original related practice questions, not additional verified NEET PYQs. In each, find the second planet’s surface escape speed divided by the first.

What if equal-mass planets have a fourfold radius difference?

Original related practice 1: The second escape speed is half the first. The planets have equal mass, and the second radius is four times the first. $R_{2} = 4 R_{1}$

\frac{v_{2}}{v_{1}} = \sqrt{\frac{R_{1}}{R_{2}}} = \sqrt{\frac{1}{4}} = \frac{1}{2}

The larger radius lowers surface escape speed. The square root turns the radius factor of four into a speed factor of two.

What if the second planet has four times the mass and twice the radius?

Original related practice 2: The second escape speed is the square root of two times the first. Equal-mass cancellation is no longer allowed.

M_{2} = 4 M_{1}, R_{2} = 2 R_{1}

\frac{v_{2}}{v_{1}} = \sqrt{\frac{M_{2}}{M_{1}} \frac{R_{1}}{R_{2}}} = \sqrt{4 \times \frac{1}{2}} = \sqrt{2}

Keep the mass ratio in the same order as the speeds. Reverse the radius ratio.

What if equal-density planets have radii differing by a factor of two?

Original related practice 3: The second escape speed is half the first. The spherical planets have equal mean density, but the second has half the radius. Equal density does not mean equal mass: mass scales with radius cubed.

M = \frac{4}{3} π ρ R^{3}, R_{2} = \frac{R_{1}}{2}

\frac{M_{2}}{M_{1}} = {(\frac{R_{2}}{R_{1}})}^{3} = \frac{1}{8}

\frac{v_{2}}{v_{1}} = \sqrt{\frac{1}{8} \times 2} = \frac{1}{2}

Compare the half-radius results:

Equal mass: the speed ratio is $\sqrt{2}$
Equal density: the speed ratio is $\frac{1}{2}$

The answers differ because the masses no longer match. Before substituting, identify what stays constant, then attempt Gravitation Practice Questions NEET: 6 Worked MCQs.

Frequently asked questions

What is the correct answer to the escape speed NEET 2026 question?

Option C, √2, is correct. For two equally massive planets with R₂ = R₁/2, the surface escape-speed ratio is v₂/v₁ = √(R₁/R₂) = √2.

What formula should I use for escape-speed ratio questions?

Surface escape speed is v = √(2GM/R), neglecting atmospheric resistance and rotation in the isolated-planet model. For two planets, v₂/v₁ = √[(M₂/M₁)(R₁/R₂)]. Cancel the mass ratio only when the masses are equal.

Why does halving the radius not double escape speed?

At fixed mass, escape speed is proportional to R⁻¹ᐟ², not R⁻¹. Halving the radius doubles the square of escape speed, so the speed itself increases by √2, not 2.

How does escape speed change with radius for equal-density planets?

For spherical planets of equal mean density, mass is proportional to radius cubed. Substituting this into v = √(2GM/R) gives v proportional to R. Therefore, halving the radius halves the surface escape speed.

Escape Speed NEET 2026: Half-Radius Ratio Explained

What is the correct answer to the escape speed NEET 2026 question?

How do you write the escape-speed formula for both planets?

How do you substitute the radius and check the answer?

Why does option D come from a method error?

How do you solve three related escape-speed ratio questions?

What if equal-mass planets have a fourfold radius difference?

What if the second planet has four times the mass and twice the radius?

What if equal-density planets have radii differing by a factor of two?

Frequently asked questions

What is the correct answer to the escape speed NEET 2026 question?

What formula should I use for escape-speed ratio questions?

Why does halving the radius not double escape speed?

How does escape speed change with radius for equal-density planets?

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