Three planets of same density have radii `R_(1),R_(2)` and `R_(3)` such that `R_(1) = 2R_(2) = 3R_(3)`. The gravitational field at their respective surfaces are `g_(1), g_(2)` and `g_(3)` and escape velocities from their surfeces are `upsilon_(1),upsilon_(2)` and `upsilon_(3)`, then

Options

A. `g_(1)//g_(2) = 2`
B. `g_(1)//g_(3) = 3`
C. `upsilon_(1)//upsilon_(2) = 1//4`
D. `upsilon_(1)//upsilon_(3) = 3`

Explanation

To solve the problem, we need to derive the relationships between the gravitational fields and escape velocities of the three planets based on their radii.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Radii:**
Given that \( R_1 = 2R_2 = 3R_3 \), we can express the radii in terms of \( R_3 \):
\[ R_1 = 6R_3, \quad R_2 = 3R_3, \quad R_3 = R_3 \]

2. **Gravitational Field at the Surface of a Planet:**
The gravitational field \( g \) at the surface of a planet is given by:
\[ g = \frac{GM}{R^2} \]
where \( M \) is the mass of the planet and \( R \) is its radius.

3. **Mass of the Planet:**
The mass \( M \) can be expressed in terms of density \( \rho \) and volume \( V \):
\[ M = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]
Substituting this into the equation for \( g \):
\[ g = \frac{G \left(\rho \frac{4}{3} \pi R^3\right)}{R^2} = \frac{4\pi G \rho}{3} R \]
This shows that \( g \) is directly proportional to \( R \).

4. **Calculating Gravitational Fields:**
Using the relationship derived, we can express the gravitational fields for the three planets:
\[ g_1 = \frac{4\pi G \rho}{3} R_1, \quad g_2 = \frac{4\pi G \rho}{3} R_2, \quad g_3 = \frac{4\pi G \rho}{3} R_3 \]
Now substituting the values of \( R_1 \), \( R_2 \), and \( R_3 \):
\[ g_1 = \frac{4\pi G \rho}{3} (6R_3) = 8g_3, \quad g_2 = \frac{4\pi G \rho}{3} (3R_3) = 4g_3 \]

5. **Finding Ratios of Gravitational Fields:**
Now we can find the ratios:
\[ \frac{g_1}{g_2} = \frac{8g_3}{4g_3} = 2 \quad \text{and} \quad \frac{g_1}{g_3} = \frac{8g_3}{g_3} = 8 \]

6. **Escape Velocity Formula:**
The escape velocity \( v \) from the surface of a planet is given by:
\[ v = \sqrt{2gR} \]
Substituting for each planet:
\[ v_1 = \sqrt{2g_1R_1}, \quad v_2 = \sqrt{2g_2R_2}, \quad v_3 = \sqrt{2g_3R_3} \]

7. **Calculating Escape Velocities:**
Using the gravitational fields we derived:
\[ v_1 = \sqrt{2 \cdot 8g_3 \cdot 6R_3} = \sqrt{96g_3R_3}, \quad v_2 = \sqrt{2 \cdot 4g_3 \cdot 3R_3} = \sqrt{24g_3R_3}, \quad v_3 = \sqrt{2g_3R_3} \]

8. **Finding Ratios of Escape Velocities:**
Now we can find the ratios:
\[ \frac{v_1}{v_2} = \frac{\sqrt{96g_3R_3}}{\sqrt{24g_3R_3}} = \sqrt{4} = 2 \]
\[ \frac{v_1}{v_3} = \frac{\sqrt{96g_3R_3}}{\sqrt{2g_3R_3}} = \sqrt{48} = 4\sqrt{3} \]

### Summary of Results:
- \( \frac{g_1}{g_2} = 2 \)
- \( \frac{g_1}{g_3} = 8 \)
- \( \frac{v_1}{v_2} = 2 \)
- \( \frac{v_1}{v_3} = 4\sqrt{3} \)