Question

The ratio of energy required to raise a satellite of mass 'm' to a height 'h' above the earth's surface of that required to put it into the orbit at same height is ______.

[R = radius of the earth]

विकल्प

• `"h"/"R"`

• `"3h"/"R"`

• `"4h"/"R"`

• `"2h"/"R"`

Answer 1

The ratio of energy required to raise a satellite of mass 'm' to a height 'h' above the earth's surface of that required to put it into the orbit at same height is `underline("2h"/"R")`.

Explanation:

The energy required to raise a satellite of mass m to a height h,

`"E"_1 = "mgh"/(1 + "h"/"R")`  ...(i)

The energy required to put it into orbit,

`"E"_2 = 1/2"mv"_0^2 = 1/2"m"("GM"/"r")  ...(because "orbital velocity", "v"_0 = "GM"/"r")`

`= 1/2"m"("GM"/("R + h")) = 1/2 "mgR"^2/("R"(1 + "h"/"R"))   ...(because "GM" = "gR"^2)`

`= "mgR"/(2 (1 + "h"/"R"))`

The ratio of E₁ : E₂ becomes,

`"E"_1/"E"_2 = "mgh"/(1 + "h"/"R") xx (2(1 + "h"/"R"))/"mgR" = "2h"/"R"`