The ratio of the masses and radii of two planets are 2:3 and 8:27.The ratio of respective escape speeds from their surfaces are

Options

A. `sqrt3:sqrt2`
B. 9 : 4`
C. 3 : 2`
D. 3 : 4`

Answer

To find the ratio of the escape speeds of two planets given the ratios of their masses and radii, we can follow these steps:

### Step 1: Understand the formula for escape speed
The escape speed \( v_e \) 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: Establish the relationship for two planets
For two planets, we can express the escape speeds as:
\[ v_{e1} = \sqrt{\frac{2GM_1}{R_1}} \quad \text{and} \quad v_{e2} = \sqrt{\frac{2GM_2}{R_2}} \]

### Step 3: Find the ratio of escape speeds
The ratio of the escape speeds \( \frac{v_{e1}}{v_{e2}} \) can be expressed as:
\[ \frac{v_{e1}}{v_{e2}} = \frac{\sqrt{\frac{2GM_1}{R_1}}}{\sqrt{\frac{2GM_2}{R_2}}} = \sqrt{\frac{M_1}{R_1} \cdot \frac{R_2}{M_2}} \]

### Step 4: Substitute the given ratios
Given:
- The ratio of the masses \( \frac{M_1}{M_2} = \frac{2}{3} \)
- The ratio of the radii \( \frac{R_1}{R_2} = \frac{8}{27} \)

We can rewrite \( \frac{R_2}{R_1} \) as:
\[ \frac{R_2}{R_1} = \frac{27}{8} \]

### Step 5: Substitute into the escape speed ratio
Now substituting the values into the escape speed ratio:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{M_1}{M_2} \cdot \frac{R_2}{R_1}} = \sqrt{\frac{2}{3} \cdot \frac{27}{8}} \]

### Step 6: Calculate the expression
Calculating the expression:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{2 \cdot 27}{3 \cdot 8}} = \sqrt{\frac{54}{24}} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]

### Conclusion
Thus, the ratio of the escape speeds from their surfaces is:
\[ \frac{v_{e1}}{v_{e2}} = \frac{3}{2} \]