Solution:

Given, height of drop = 2Re

Let's assume that the particles A and B are dropped from two points on earth's surface, which are separated by a distance of d.

To find the separation between particles A and B, we need to calculate the time taken by each of them to reach the earth's surface.

Time taken to reach earth's surface:

The time taken by a particle to reach the earth's surface can be calculated using the formula:

t = √(2h/g)

where,
h = height of drop (2Re)
g = acceleration due to gravity (9.8 m/s^2)

t = √(2(2Re)/9.8)
= √(4Re/4.9)
= √(4Re/4.9) * √(4.9/4.9) [multiplying and dividing by 4.9]
= √(19.6Re/24.01)
= √(0.816Re)
= 0.903Re^(1/2)

Thus, the time taken by particles A and B to reach the earth's surface will be:

t(A) = t(B) = 0.903Re^(1/2)

Separation between particles A and B:

The horizontal separation between particles A and B can be calculated using the formula:

s = vt

where,
v = horizontal velocity of the particles (which is constant and equal to the velocity of earth's rotation at that point)
t = time taken by the particles to reach the earth's surface (which is same for both particles)

The velocity of earth's rotation at a point can be calculated using the formula:

v = ωr

where,
ω = angular velocity of earth's rotation
r = distance of the point from earth's axis of rotation

The angular velocity of earth's rotation is given by:

ω = 2π/T

where,
T = time period of earth's rotation (24 hours)

The distance of the point from earth's axis of rotation can be calculated using the formula:

r = Re * cos(θ)

where,
Re = radius of earth
θ = latitude of the point

Since the particles are dropped from two points which are slightly away from one another, their latitudes will be slightly different. Therefore, their distances from the axis of rotation will also be different.

Let's assume that the latitude of particle A is θ1 and the latitude of particle B is θ2. Then, their distances from the axis of rotation will be:

r(A) = Re * cos(θ1)
r(B) = Re * cos(θ2)

The horizontal separation between particles A and B will be:

s = v * t
= ωr * t
= 2π/T * [(Re * cos(θ1)) + (Re * cos(θ2))] * 0.903Re^(1/2)
= 2π * Re * (cos(θ1) + cos(θ2)) * 0.903Re^(1/2)

Therefore, the separation between particles A and B is given by:

s = 2π * Re * (cos(θ1) + cos(θ2)) * 0.903Re^(1/2)

Note: Since the separation between particles A and B depends