Question

The ratio of the radii of the planets P₁​ and P₂​ is k. The ratio of acceleration due to gravity is r. The ratio of the escape velocities from them will be ______.

विकल्प

• kr

• `sqrt"kr"`

• `sqrt("k"/"r")`

• `sqrt("r"/"k")`

Answer 1

The ratio of the radii of the planets P₁​ and P₂​ is k. The ratio of acceleration due to gravity is r. The ratio of the escape velocities from them will be `underlinebb(sqrt"kr")`.

Explanation:

We are given that:

`("GM"_1/"R"_1^2)/("GM"_2/"R"_2^2)="r"`

`"R"_1/"R"_2` = k

And the escape velocity is given by: v = `sqrt((2"GM")/"R")` where M is the mass of the planet and R is its radius.

So, `"v"_1/"v"_2=sqrt("M"_1/"M"_2//"R"_1/"R"_2)`

= `sqrt("rk"^2/"k")`

= `sqrt"kr"`