Q. 68

Expert-verifiedFound in: Page 356

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + Δr, where Δr << r , has the very slightly different period T+ ΔT.

a) Show that

b) Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

a) The given expression is proved.

b) They will meet again after 281.05 days

Two satellites are crossing each other. r and T are the radius of orbit and time period of first satellite whereas r + Δr, and T + ΔT, are the radius of orbit and time period of the second satellite respectively.

From Kepler's law

Now lets assume , the first satellite obeys the .

And , for the second satellite as it varies with Δr and ΔT so it will be

as

so,

Now subtracting the equation for the first satellite:

On dividing with the equation for T, we get

And proved.

Two satellites are in parallel orbit.

Radius of orbit of first satellite,r=6700 kmRadius of orbit of second satellite, r + Δr=6701 kmMass of Earth =5.98 x 10^{24} kg

In part(a) of the problem we have established

Substitute values

So after periods, they will meet again.

Find the value of One period

So they will meet again after 4467 periods = 4467 x 1.51 hours =6745.2 hours

.6745.2 hours =281.05 days .

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