CBSE Class 12-science Answered

It is understood that required velocity at infinity is 15 km/s, what is projeected velocity at earth's surface.

Let m be the mass of body .

Total energy E_o on earth surface is

Where first term of RHS is potential energy and second term is kinetic energy .

Where G is gravitational potential energy , M is mass of earth , m is mass of body , r is radius of earth and

v_o is velocity of body on earth's surface.

At infinity , potential energy is zero and the body has only kinetic energy .

Hence energy at infinity is

where v_∞ is velocity at infinity.

By conservation of energy , 

E_o = E_∞

if acceleration due to gravity  g = G (M/r²) , then above expression is simplifield for v_o as

Let us substitute  g = 9.8 m/s² , r = 6.371 × 10⁶ m ,  v_∞ = 15 × 10³ m/s in above expression to calculate v_o .

we get v_o = 18.70 m/s