The escape velocities on the surface of two planets of masses `m_1` and `m_2` and having the radius `r_1` and `r_2` are `v_1` and `v_2` respectively, then

Options

A. `v_1/v_2=(m_1r_1)/(m_2r_2)`
B. `v_2/v_1=m_1/m_2 r_1/r_2`
C. `v_2/v_1=sqrt(m_2/m_1 r_1/r_2)`
D. `v_2/v_1=sqrt((m_1r_1)/(m_2r_2))`

Answer

To find the ratio of escape velocities \( v_1 \) and \( v_2 \) for two planets with masses \( m_1 \) and \( m_2 \), and radii \( r_1 \) and \( r_2 \), we will use the formula for escape velocity.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( v \) from the surface of a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Escape Velocity for Planet 1**:
For planet 1, we have:
\[ v_1 = \sqrt{\frac{2Gm_1}{r_1}} \]

3. **Escape Velocity for Planet 2**:
For planet 2, we have:
\[ v_2 = \sqrt{\frac{2Gm_2}{r_2}} \]

4. **Finding the Ratio of Escape Velocities**:
To find the ratio \( \frac{v_1}{v_2} \), we can write:
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{2Gm_1}{r_1}}}{\sqrt{\frac{2Gm_2}{r_2}}} \]

5. **Simplifying the Ratio**:
This simplifies to:
\[ \frac{v_1}{v_2} = \sqrt{\frac{m_1}{m_2} \cdot \frac{r_2}{r_1}} \]

6. **Final Expression**:
Therefore, the ratio of the escape velocities can be expressed as:
\[ \frac{v_1}{v_2} = \sqrt{\frac{m_1 \cdot r_2}{m_2 \cdot r_1}} \]

### Conclusion:
The final result shows how the escape velocities of the two planets relate to their masses and radii.