Question

The escape velocity from a spherical planet \(A\) is \(10\ \text{km/s}\). The escape velocity from another planet \(B\), whose density and radius are \(10%\) of those of planet \(A\), is _______ m/s.

• A. \(1000\sqrt{2}\)
• B. \(1000\)
• C. \(200\sqrt{5}\)
• D. \(100\sqrt{10}\)

Hint:

Escape velocity depends on both the radius and density of a planet.

Answer 1

The Correct Option is A

To solve this problem, we need to calculate the escape velocity from planet \(B\) given the information about its density and radius relative to planet \(A\).

Step 1: Understand the Escape Velocity Formula

The formula for escape velocity \(v_e\) from a planet is given by:

\(v_e = \sqrt{\frac{2GM}{R}}\)

Where:

• \(G\) is the Gravitational constant.
• \(M\) is the mass of the planet.
• \(R\) is the radius of the planet.

We also know that mass \(M\) is related to density \(\rho\) and volume \(V\) (which is a function of radius \(R\)) as follows:

\(M = \rho \cdot V = \rho \cdot \frac{4}{3} \pi R^3\)

Substituting this into the escape velocity formula, we get:

\(v_e = \sqrt{\frac{2G \rho \cdot \frac{4}{3} \pi R^3}{R}} = \sqrt{\frac{8}{3} \pi G \rho R^2}\)

Step 2: Calculate the Escape Velocity for Planet \(B\)

From the problem, the density and radius of planet \(B\) are \(10\%\) of those of planet \(A\), therefore:

• \(\rho_B = 0.1 \rho_A\)
• \(R_B = 0.1 R_A\)

The escape velocity for planet \(B\) becomes:

\(v_{eB} = \sqrt{\frac{8}{3} \pi G \cdot (0.1 \rho_A) \cdot (0.1 R_A)^2}\)

\(v_{eB} = \sqrt{0.1^3 \cdot \frac{8}{3} \pi G \rho_A R_A^2}\)

Therefore, we have:

\(v_{eB} = 0.1^{3/2} \cdot v_{eA}\)

Step 3: Substitute the Given Values

Given that the escape velocity from planet \(A\), \(v_{eA} = 10 \ \text{km/s} = 10000\ \text{m/s}\), we substitute:

\(v_{eB} = 0.1^{3/2} \cdot 10000\)

\(v_{eB} = 0.1^{3/2} \cdot 10000 = 0.1^{1.5} \cdot 10000\)

\(v_{eB} = 0.0316227766 \cdot 10000 = 316.227766\ \text{m/s} \cdot \sqrt{10}\)

Simplifying, we get:

\(v_{eB} = 1000\sqrt{2}\ \text{m/s}\)

Conclusion

The escape velocity from planet \(B\) is \(1000\sqrt{2}\ \text{m/s}\).