Question

A ball 'a' of mass m falls to the surface of the earth from infinity. Another ball 'b' of mass 2m falls to the earth from the height equal to six times radius of the earth then ratio of velocities of 'a' and 'b' on reaching the earth is

Answer 1

✿.｡.:* ☆.:* soⓁᵘｉᶰ *:.☆*.:｡.✿

As we know the concept of potential energy.

So I don't wanna make u confuse so only one point that in this case I assume potential energy of the system ( Earth and ball) Zero at infinity.

✿.｡.:* ☆:**:. MATHEMATICALLY .:**:.☆*.:｡.✿

Potential energy of the system (Earth and ball) at distance of s can be written as....

U = -GMm/s

G is gravitation constant

M mass of earth

m is mass of balls.

For ball "A"

Let the initial velocity be zero as u not mention.

Final velocity be V

Initial potential energy = 0 discussed above

Final potential energy = -GMm/R

ACC to conservation of energy....

1/2mv² - GMm/R = 0

1/2mv² = GMm/R

v² = 2GM/R

v = (2GM/R)^½

For ball "B"

Initial speed = 0

Initial potential energy = -GMm'/6R

Final speed u

Final potential energy = -GMm'/R

1/2m'u² - GMm'/R = -GMm'/6R

1/2m'u² = GMm'/R - GMm'/6R

1/2u² = (6GM-GM)/6R

u² = (5GM)/3R

u = (5GM/3R)^½

RATIO = V : U

RATIO = (2GM/R)^½ : (5GM/3R)^½

RATIO = √6 : √5

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