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 from the surfaces of two planets, we can use the formula for escape speed:

\[ V_e = \sqrt{\frac{2GM}{R}} \]

where:
- \( V_e \) is the escape speed,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

Given:
- The ratio of the masses of the two planets \( \frac{M_1}{M_2} = \frac{2}{3} \)
- The ratio of the radii of the two planets \( \frac{R_1}{R_2} = \frac{8}{27} \)

We need to find the ratio of the escape speeds \( \frac{V_1}{V_2} \).

### Step 1: Write the escape speed formulas for both planets

For planet 1:
\[ V_1 = \sqrt{\frac{2GM_1}{R_1}} \]

For planet 2:
\[ V_2 = \sqrt{\frac{2GM_2}{R_2}} \]

### Step 2: Find the ratio of escape speeds

Now, we can find the ratio \( \frac{V_1}{V_2} \):

\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{2GM_1}{R_1}}}{\sqrt{\frac{2GM_2}{R_2}}} \]

This simplifies to:

\[ \frac{V_1}{V_2} = \sqrt{\frac{M_1}{M_2} \cdot \frac{R_2}{R_1}} \]

### Step 3: Substitute the given ratios

Substituting the given ratios into the equation:

\[ \frac{V_1}{V_2} = \sqrt{\frac{2/3}{8/27}} \]

### Step 4: Simplify the expression

To simplify \( \frac{2/3}{8/27} \):

\[ \frac{2}{3} \cdot \frac{27}{8} = \frac{2 \cdot 27}{3 \cdot 8} = \frac{54}{24} = \frac{9}{4} \]

### Step 5: Take the square root

Now, taking the square root:

\[ \frac{V_1}{V_2} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]

### Conclusion

Thus, the ratio of the escape speeds from the surfaces of the two planets is:

\[ \frac{V_1}{V_2} = \frac{3}{2} \]

### Final Answer

The ratio of respective escape speeds from their surfaces is \( 3:2 \).

---