The average radii of orbits of mercury and earth around the sun are `6xx10^(7)` km and `1.5xx10^(8)` km respectively. The ratio of their orbital speeds will be :-

Options

A. `sqrt(5) : sqrt(2)`
B. `sqrt(2) : sqrt(5)`
C. `2.5 :1`
D. `1 : 25`

Answer

To find the ratio of the orbital speeds of Mercury and Earth around the Sun, we can follow these steps:

### Step 1: Understand the relationship between orbital speed and radius
The orbital speed \( v \) of a planet is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Sun, and \( r \) is the radius of the orbit. From this formula, we can see that the orbital speed is inversely proportional to the square root of the radius of the orbit.

### Step 2: Set up the ratio of the orbital speeds
Let \( v_M \) be the orbital speed of Mercury and \( v_E \) be the orbital speed of Earth. The ratio of their speeds can be expressed as:
\[ \frac{v_M}{v_E} = \sqrt{\frac{r_E}{r_M}} \]
where \( r_E \) is the radius of Earth's orbit and \( r_M \) is the radius of Mercury's orbit.

### Step 3: Substitute the given values
The average radii of the orbits are:
- \( r_M = 6 \times 10^7 \) km (for Mercury)
- \( r_E = 1.5 \times 10^8 \) km (for Earth)

Now, substituting these values into the equation:
\[ \frac{v_M}{v_E} = \sqrt{\frac{1.5 \times 10^8}{6 \times 10^7}} \]

### Step 4: Simplify the ratio
We can simplify the fraction inside the square root:
\[ \frac{1.5 \times 10^8}{6 \times 10^7} = \frac{1.5}{6} \times \frac{10^8}{10^7} = \frac{1.5}{6} \times 10^{1} = \frac{1.5}{6} \times 10 = \frac{15}{60} \times 10 = \frac{1}{4} \times 10 = \frac{10}{4} = 2.5 \]
Thus, we have:
\[ \frac{v_M}{v_E} = \sqrt{2.5} \]

### Step 5: Express in simplest form
We can express \( 2.5 \) as:
\[ 2.5 = \frac{5}{2} \]
So,
\[ \frac{v_M}{v_E} = \sqrt{\frac{5}{2}} = \frac{\sqrt{5}}{\sqrt{2}} \]

### Step 6: Final ratio
Thus, the ratio of the orbital speeds of Mercury to Earth is:
\[ \frac{v_M}{v_E} = \sqrt{5} : \sqrt{2} \]

### Conclusion
The final answer is:
\[ \text{The ratio of the orbital speeds of Mercury and Earth is } \sqrt{5} : \sqrt{2}. \]

---