Two planets of same density have the ratio of their radii as 1 : 3. The ratio of escape speed on them will be

Options

A. `9 : 1`
B. `1 : 9`
C. `1 : 3`
D. `3 : 1`

Answer

To solve the problem, we need to find the ratio of escape speeds of two planets that have the same density and a radius ratio of 1:3. Let's break this down step by step.

### Step 1: Understand the formula for escape velocity
The escape velocity \( v_e \) from the surface of a planet is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 2: Express mass in terms of density and volume
The mass \( M \) of a planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V \]
For a spherical planet, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we can write:
\[ M = \rho \left(\frac{4}{3} \pi R^3\right) \]

### Step 3: Substitute mass in the escape velocity formula
Substituting the expression for mass \( M \) into the escape velocity formula gives:
\[ v_e = \sqrt{\frac{2G \left(\rho \frac{4}{3} \pi R^3\right)}{R}} \]
This simplifies to:
\[ v_e = \sqrt{\frac{8\pi G \rho R^2}{3}} \]

### Step 4: Analyze the escape velocity
From the above expression, we can see that the escape velocity \( v_e \) is directly proportional to the square root of the radius \( R \):
\[ v_e \propto \sqrt{R} \]

### Step 5: Find the ratio of escape speeds
Let the radii of the two planets be \( R_1 \) and \( R_2 \) with the ratio \( R_1 : R_2 = 1 : 3 \). Therefore, we can write:
\[ R_1 = R \quad \text{and} \quad R_2 = 3R \]

Now, we can find the ratio of their escape speeds:
\[ \frac{v_{e1}}{v_{e2}} = \frac{\sqrt{R_1}}{\sqrt{R_2}} = \frac{\sqrt{R}}{\sqrt{3R}} = \frac{1}{\sqrt{3}} \]

### Step 6: Conclusion
Thus, the ratio of escape speeds \( v_{e1} : v_{e2} \) is:
\[ v_{e1} : v_{e2} = 1 : \sqrt{3} \]

### Final Answer
The ratio of escape speed on the two planets is \( 1 : \sqrt{3} \).

---