The spherical planets have the same mass but densities in the ratio `1: 8`. For these planets the :

Options

A. acceleration due to gravity will be in the ratio `4 : 1`
B. acceleration due to gravity will be in the ratio `1 : 4`
C. escape velocities from their surfaces will be `i` the ratio `sqrt(2) : 1`
D. escape velocities from their surfaces will be in the ratio `1 : sqrt(2)`

Answer

To solve the problem, we need to analyze the relationship between the densities, volumes, radii, gravitational accelerations, and escape velocities of two spherical planets with the same mass but different densities.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The planets have the same mass (let's denote it as \( M \)).
- The densities of the planets are in the ratio \( \rho_1 : \rho_2 = 1 : 8 \).

2. **Using the Density and Volume Relationship**:
- The mass of a sphere can be expressed as:
\[ M = \rho V \]
- For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Therefore, we can write the mass for each planet as:
\[ M_1 = \rho_1 \left(\frac{4}{3} \pi r_1^3\right) \quad \text{and} \quad M_2 = \rho_2 \left(\frac{4}{3} \pi r_2^3\right) \]

3. **Setting the Masses Equal**:
- Since the masses are equal, we have:
\[ \rho_1 \left(\frac{4}{3} \pi r_1^3\right) = \rho_2 \left(\frac{4}{3} \pi r_2^3\right) \]
- The \( \frac{4}{3} \pi \) cancels out:
\[ \rho_1 r_1^3 = \rho_2 r_2^3 \]

4. **Substituting the Density Ratio**:
- Given \( \rho_1 : \rho_2 = 1 : 8 \), we can express this as:
\[ \rho_1 = 1 \quad \text{and} \quad \rho_2 = 8 \]
- Substituting these values into the equation gives:
\[ 1 \cdot r_1^3 = 8 \cdot r_2^3 \]
- Rearranging this, we find:
\[ r_1^3 = 8 r_2^3 \]
- Taking the cube root of both sides:
\[ r_1 = 2 r_2 \]
- This means the ratio of the radii is:
\[ \frac{r_1}{r_2} = 2 : 1 \]

5. **Finding the Gravitational Acceleration**:
- The formula for gravitational acceleration \( g \) at the surface of a planet is given by:
\[ g = \frac{GM}{R^2} \]
- Since the mass \( M \) is the same for both planets, we can express the ratio of gravitational accelerations as:
\[ \frac{g_1}{g_2} = \frac{M/r_1^2}{M/r_2^2} = \frac{r_2^2}{r_1^2} \]
- Substituting \( r_1 = 2 r_2 \):
\[ \frac{g_1}{g_2} = \frac{r_2^2}{(2 r_2)^2} = \frac{r_2^2}{4 r_2^2} = \frac{1}{4} \]
- Thus, the ratio of gravitational accelerations is:
\[ g_1 : g_2 = 1 : 4 \]

6. **Finding the Escape Velocity**:
- The escape velocity \( v_e \) is given by:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
- The ratio of escape velocities is:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{R_2}{R_1}} \]
- Substituting \( R_1 = 2 R_2 \):
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{R_2}{2R_2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]
- Thus, the ratio of escape velocities is:
\[ v_{e1} : v_{e2} = 1 : \sqrt{2} \]

### Final Results:
- The ratio of gravitational accelerations \( g_1 : g_2 = 1 : 4 \).
- The ratio of escape velocities \( v_{e1} : v_{e2} = 1 : \sqrt{2} \).