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nound a Star a verocity 4x 1ou can conewe to the Stardas eto the Star as Pa seto the Staras Pg away from the Sta

nound a Star a verocity 4x 1ou can conewe to the Stardas eto the Star as Pa seto the Staras Pg away from the Sta

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nound a Star a verocity 4x 1ou can conewe to the Stardas eto the Star as Pa seto the Staras Pg away from the Sta
Transcribed Image Text:nound a Star a verocity 4x 1ou can conewe to the Stardas eto the Star as Pa seto the Staras Pg away from the Sta
Expert Solution
Step 1

Newton's Law of Gravitation

According to Newton's Law of gravitation, the gravitational force between two objects of masses m1 and m2 with the distance between them r is given by

F=Gm1m2r2

Let us consider that the orbit of mass m2 around m1 is a circle. To remain in the orbit, the gravitational force must balance the centripetal force. If v is the velocity of mass m2 then

m2v2r=Gm1m2r2

Thus the velocity of the mass m2 around m1 is given by

v=Gm1r

Thus we see that the velocity of the body is independent of its mass and depends only on the distance between them.

Step 2

Let the mass around which the planets P1 and P2 rotate be M. Let their velocities be v1 and v2 and the distance between them and the star be R1 and R2 respectively. Therefore we have

v1=GMR1v2=GMR2

In the question, it is given that the velocity of the planet P1 is four times the velocity of the planet P2. Thus

v1=4v2

Now taking ratio

v1v2=GMR1GMR2=R2R1⇒4v2v2=R2R1

This gives

R2R1=42⇒R2=16R1⇒R1=116R2

This says that P1 is 116 times closer to the star than P2.

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