The escape velocities on the surface of two planets of masses `m_1` and `m_2` and having same radius `v_1` and `v_2` respectively . Then

Options

A. `v_1/v_2=sqrt(m_1/m_2)`
B. `v_2/v_1=m_1/m_2`
C. `v_1/v_2=sqrt(m_2/m_1)`
D. `v_1/v_2=(m_2/m_1)^2`

Answer

To solve the problem of finding the ratio of escape velocities \( v_1 \) and \( v_2 \) on the surfaces of two planets with masses \( m_1 \) and \( m_2 \) and the same radius \( r \), we can follow these steps:

### Step-by-Step Solution:

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.

2. **Write the Escape Velocity for Each Planet**:
For planet 1 with mass \( m_1 \) and radius \( r \):
\[ v_1 = \sqrt{\frac{2Gm_1}{r}} \]
For planet 2 with mass \( m_2 \) and radius \( r \):
\[ v_2 = \sqrt{\frac{2Gm_2}{r}} \]

3. **Set Up the Ratio of Escape Velocities**:
We want to find the ratio \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{2Gm_1}{r}}}{\sqrt{\frac{2Gm_2}{r}}} \]

4. **Simplify the Ratio**:
The \( \sqrt{2G} \) and \( \sqrt{r} \) terms cancel out:
\[ \frac{v_1}{v_2} = \frac{\sqrt{m_1}}{\sqrt{m_2}} = \sqrt{\frac{m_1}{m_2}} \]

5. **Final Result**:
Thus, the ratio of the escape velocities is:
\[ \frac{v_1}{v_2} = \sqrt{\frac{m_1}{m_2}} \]