Question

A planet 'A' having density \( \rho \) and radius \( R \) has escape velocity \( = 10 \, \text{km/sec} \). Find the escape velocity of a planet B having density and radius both 10% that of planet A.

• A. \( \frac{1}{\sqrt{10}} \)
• B. \( \frac{1}{\sqrt{20}} \)
• C. \( \frac{1}{\sqrt{30}} \)
• D. \( \frac{1}{\sqrt{50}} \)

Hint:

The escape velocity depends on the square root of the mass-to-radius ratio, so changes in radius and density affect the velocity significantly.

Answer 1

The Correct Option is A

To find the escape velocity of planet B, we first recall the expression for escape velocity from a planet.

v_e = √(2GM / R)

where:

• G is the gravitational constant,
• M is the mass of the planet,
• R is the radius of the planet.

The mass of a spherical planet can be written in terms of its density and radius as:

M = ρ × (4/3)πR³

Substituting this expression for mass into the escape velocity formula:

v_e = √(2G × (4/3)πρR³ / R)

v_e = √(8πGρR² / 3)

Thus, escape velocity depends on the square root of density and directly on the radius.

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Comparison Between Planets A and B

Given:

• Escape velocity of planet A, v_eA = 10 km/s
• Density of planet B = 0.1 × density of planet A
• Radius of planet B = 0.1 × radius of planet A

Using the relation v_e ∝ √ρ × R, we write:

v_eB / v_eA = √(ρ_B / ρ_A) × (R_B / R_A)

Substituting the given ratios:

v_eB / v_eA = √(0.1) × 0.1 = √(0.001)

Hence:

v_eB = √(0.001) × 10 km/s

v_eB = 1 / √10 km/s

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Final Answer:

Escape velocity of planet B = 1 / √10 km/s