Two artificial satellites a and b are at a distances ra and rb above the earth’s surface. If the radius of earth is R, then the ratio of their speeds will be

The correct option is [(rb + R)/(ra + R)]⁰.⁵.

Let us derive the ratio of the speeds of the two artificial satellites step-by-step.

Step 1: Understand the formula for orbital velocity
The orbital speed $v$ of a satellite orbiting the Earth at a distance $r$ from the center of the Earth is determined by the gravitational force acting as the centripetal force:
$\frac{m{v}^2}{r}=\frac{GMm}{r^2}$
Solving for $v$, we get the orbital speed formula:
$v=\sqrt{\frac{GM}{r}}$
where:
- $G$ is the universal gravitational constant,
- $M$ is the mass of the Earth,
- $r$ is the orbital radius (distance from the center of the Earth to the satellite).

Step 2: Determine the orbital radii of the satellites
The distance of a satellite from the center of the Earth is the sum of the Earth's radius ($R$) and the altitude of the satellite above the Earth's surface.
For satellite a, the altitude is $r_a$. Therefore, its orbital radius is:
$r_{\text{orbit, a}}=r_a+R$
For satellite b, the altitude is $r_b$. Therefore, its orbital radius is:
$r_{\text{orbit, b}}=r_b+R$

Step 3: Calculate the orbital speeds of both satellites
Using the orbital velocity formula, the speed of satellite a ($v_a$) is:
$v_a=\sqrt{\frac{GM1}{r_a+R}}$
Similarly, the speed of satellite b ($v_b$) is:
$v_b=\sqrt{\frac{GM}{r_b+R}}$

Step 4: Find the ratio of their orbital speeds
Now, we take the ratio of the speed of satellite a to the speed of satellite b:
$\frac{v_a}{v_b}=\frac{\sqrt{\frac{GM}{r_a+R}}}{\sqrt{\frac{GM}{r_b+R}}}$
Since $G$ and $M$ are constants, they cancel out, simplifying the expression to:
$\frac{v_a}{v_b}=\sqrt{\frac{r_b+R}{r_a+R}}$
We can write the square root as a fractional exponent of 0.5:
$\frac{v_a}{v_b}={\left[\frac{r_b+R}{r_a+R}\right]}^{0.5}$