Question

The escape velocity from the Earth's surface is $v$. The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is :

• A. v
• B. 2 v
• C. 3 v
• D. 4 v

Answer 1

The Correct Option is D

To solve the problem of finding the escape velocity from the surface of another planet with characteristics compared to Earth, we need to understand the formula for escape velocity and how changes in planetary characteristics affect it.

The escape velocity \(v\) from the surface of a planet is given by the formula: 

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

Where:

• \(v_e\) is the escape velocity,
• \(G\) is the universal gravitational constant,
• \(M\) is the mass of the planet,
• \(R\) is the radius of the planet.

For the Earth, the escape velocity is given as \(v\). So,

\(v = \sqrt{\dfrac{2GM_{\text{earth}}}{R_{\text{earth}}}}\)

For the other planet, it is mentioned that its radius is four times that of Earth and it has the same mass density. Let \(\rho\) be the mass density:

\(\rho = \dfrac{M_{\text{earth}}}{V_{\text{earth}}}\) and \(V = \dfrac{4}{3} \pi R^3\).

If the radius of the new planet is \(R_{\text{new}} = 4 R_{\text{earth}}\), its volume \(V_{\text{new}}\) is:

\(V_{\text{new}} = \dfrac{4}{3} \pi (4R_{\text{earth}})^3 = \dfrac{4}{3} \pi \cdot 64 R_{\text{earth}}^3\)

Therefore, the mass \(M_{\text{new}}\) is:

\(M_{\text{new}} = \rho \cdot V_{\text{new}} = \rho \cdot 64 \cdot V_\text{earth} = 64 \cdot M_{\text{earth}}\)

Now, substitute \(M_{\text{new}}\) and \(R_{\text{new}}\) back into the escape velocity equation to find the escape velocity for the new planet:

\(v_{\text{new}} = \sqrt{\dfrac{2G \cdot 64 \cdot M_{\text{earth}}}{4 \cdot R_{\text{earth}}}}\)

Simplifying, we get:

\(v_{\text{new}} = \sqrt{\dfrac{128G M_{\text{earth}}}{4 R_{\text{earth}}}} = \sqrt{32 \cdot \dfrac{2G M_{\text{earth}}}{R_{\text{earth}}}}\)

\(v_{\text{new}} = \sqrt{32} \cdot v = 4 \cdot v\)

Thus, the escape velocity from the surface of the new planet is \(4v\). The correct answer is 4v.