Question

Answer the following question in detail.

If the Earth were made of wood, the mass of wooden Earth would have been 10% as much as it is now (without change in its diameter). Calculate escape speed from the surface of this Earth.

Answer 1

Given: M_w = 10% of  M = `"M"/10`, D_w = D or R_w = R,

To find: Escape speed (v_e)

Formula: `"v"_e = sqrt((2"GM")/"R")`

Calculation:

From formula,

`"v"_("e"_"w") = sqrt((2"GM"_"w")/("R"_"w"))`

∴ `"v"_("e"_"w")/"v"_"e" = sqrt((2"GM"_"w")/("R"_"w") xx ("R")/(2"GM"))`

∴ `"v"_("e"_"w")/"v"_"e" = sqrt((("R")/("R"_"w")) xx (("M"_"w")/("M")))`

∴ `"v"_("e"_"w")/"v"_"e" = sqrt(1 xx 1/10) = sqrt(1/10)`

∴ `"v"_("e"_"w") = "v"_"e" xx 1/sqrt10`

As, we know that the escape speed from surface of the Earth is 11.2 km/s,

Substituting value of v_e = 11.2 km/s

`"v"_("e"_"w") = 11.2 xx 1/sqrt10 = 11.2/3.162`

`= 11.2 xx 1/3.1620`     ....[Taking square root value]

= antilog{log(11.2) - log(3.162)}

= antilog{1.0492 - 0.5000}

= antilog{0.5492}

= 3.542

∴ `"v"_("e"_"w")` = 3.54 km/s

The escape velocity from the surface of wooden Earth is 3.54 km/s.