the ratio of escape velocities of two planets if `g` value on the two planets are `9.9 m//s^(2)` and `3.3 m//s^(2)` and there are `6400 km` and `3200 km` respectively is

Options

A. `2.36:1`
B. `1.36:1`
C. `3.36:1`
D. `4.36:1`

Explanation

To find the ratio of escape velocities of two planets given their gravitational acceleration and radii, we can follow these steps:

### Step 1: Write the formula for escape velocity
The escape velocity \( v_e \) from the surface of a planet is given by the formula:
\[ v_e = \sqrt{2gr} \]
where \( g \) is the acceleration due to gravity and \( r \) is the radius of the planet.

### Step 2: Calculate escape velocity for Planet 1
For Planet 1:
- \( g_1 = 9.9 \, \text{m/s}^2 \)
- \( r_1 = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)

Using the formula:
\[ v_{e1} = \sqrt{2 \times 9.9 \times (6400 \times 10^3)} \]

### Step 3: Calculate escape velocity for Planet 2
For Planet 2:
- \( g_2 = 3.3 \, \text{m/s}^2 \)
- \( r_2 = 3200 \, \text{km} = 3200 \times 10^3 \, \text{m} \)

Using the formula:
\[ v_{e2} = \sqrt{2 \times 3.3 \times (3200 \times 10^3)} \]

### Step 4: Find the ratio of escape velocities
Now, we find the ratio of the escape velocities:
\[ \frac{v_{e1}}{v_{e2}} = \frac{\sqrt{2 \times 9.9 \times (6400 \times 10^3)}}{\sqrt{2 \times 3.3 \times (3200 \times 10^3)}} \]

### Step 5: Simplify the ratio
This simplifies to:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{9.9 \times 6400 \times 10^3}{3.3 \times 3200 \times 10^3}} \]

The \( 10^3 \) cancels out:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{9.9 \times 6400}{3.3 \times 3200}} \]

### Step 6: Further simplify the expression
We can simplify the fraction:
\[ \frac{9.9}{3.3} = 3 \quad \text{and} \quad \frac{6400}{3200} = 2 \]

So we have:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{3 \times 2} = \sqrt{6} \]

### Final Answer
Thus, the ratio of escape velocities of the two planets is:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{6} \]

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