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A satellite moves around the earth in a circular orbit of radius R centered at the earth. A second satellite moves in an elliptic orbit of major axis 8R, with the earth at one of the foci. If the former takes 1 day to complete a revolution, the latter would take
- a)21.6 days
- b)8 days
- c)3 hours
- d)1.1 hour
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A satellite moves around the earth in a circular orbit of radius R cen...
Given data:
Satellite 1:
Circular orbit of radius R
Completes one revolution in 1 day
Satellite 2:
Elliptic orbit of major axis 8R
Earth at one of the foci
To find: Time taken by Satellite 2 to complete one revolution
Explanation:
Let's assume that the distance between the earth and the foci of the elliptical orbit is 'c'.
We know that for an elliptical orbit, the time period (T) is given by:
T = 2π√(a^3/GM)
where a is the semi-major axis, G is the gravitational constant, and M is the mass of the earth.
In this case, the major axis of the elliptical orbit is 8R, so the semi-major axis (a) will be 4R.
Also, since the earth is at one of the foci, we can use the formula for the distance between the foci and the center of the ellipse:
c = √(a^2 - b^2)
where b is the semi-minor axis.
Since the orbit is circular, the radius of the circular orbit (R) is equal to the semi-major axis of the elliptical orbit (a).
Therefore, we can write:
c = √(a^2 - R^2)
c = √(16R^2 - R^2)
c = √(15R^2)
Substituting these values in the formula for time period:
T = 2π√(a^3/GM)
T = 2π√((4R)^3/GM)
T = 8π√(R^3/GM)
We know that Satellite 1 takes 1 day to complete one revolution, so its time period is 1 day or 24 hours.
Therefore, the distance traveled by Satellite 1 in one revolution is equal to the circumference of the circular orbit:
2πR
For Satellite 2, since it is traveling in an elliptical orbit, the distance traveled in one revolution will be equal to the circumference of the ellipse:
2π√((a^2 + b^2)/2)
Substituting the values of a and b:
2π√((4R)^2 + (b^2))/2
2π√(16R^2 + b^2)/2
We can use the formula for the eccentricity of an elliptical orbit to find the value of b:
e = c/a
e = √(15R^2)/(4R)
e = √15/4
Using the formula for the eccentricity, we can also find the value of b:
e = √(1 - b^2/a^2)
(√15/4)^2 = 1 - b^2/(4R)^2
15/16 = 1 - b^2/(16R^2)
b^2/(16R^2) = 1 - 15/16
b^2/(16R^2) = 1/16
b^2 = R^2
b = R
Substituting the values of a and b in the formula for distance traveled by Satellite 2:
2π√(16R^2 + R^2)/2
2π√(17R
Satellite 1:
Circular orbit of radius R
Completes one revolution in 1 day
Satellite 2:
Elliptic orbit of major axis 8R
Earth at one of the foci
To find: Time taken by Satellite 2 to complete one revolution
Explanation:
Let's assume that the distance between the earth and the foci of the elliptical orbit is 'c'.
We know that for an elliptical orbit, the time period (T) is given by:
T = 2π√(a^3/GM)
where a is the semi-major axis, G is the gravitational constant, and M is the mass of the earth.
In this case, the major axis of the elliptical orbit is 8R, so the semi-major axis (a) will be 4R.
Also, since the earth is at one of the foci, we can use the formula for the distance between the foci and the center of the ellipse:
c = √(a^2 - b^2)
where b is the semi-minor axis.
Since the orbit is circular, the radius of the circular orbit (R) is equal to the semi-major axis of the elliptical orbit (a).
Therefore, we can write:
c = √(a^2 - R^2)
c = √(16R^2 - R^2)
c = √(15R^2)
Substituting these values in the formula for time period:
T = 2π√(a^3/GM)
T = 2π√((4R)^3/GM)
T = 8π√(R^3/GM)
We know that Satellite 1 takes 1 day to complete one revolution, so its time period is 1 day or 24 hours.
Therefore, the distance traveled by Satellite 1 in one revolution is equal to the circumference of the circular orbit:
2πR
For Satellite 2, since it is traveling in an elliptical orbit, the distance traveled in one revolution will be equal to the circumference of the ellipse:
2π√((a^2 + b^2)/2)
Substituting the values of a and b:
2π√((4R)^2 + (b^2))/2
2π√(16R^2 + b^2)/2
We can use the formula for the eccentricity of an elliptical orbit to find the value of b:
e = c/a
e = √(15R^2)/(4R)
e = √15/4
Using the formula for the eccentricity, we can also find the value of b:
e = √(1 - b^2/a^2)
(√15/4)^2 = 1 - b^2/(4R)^2
15/16 = 1 - b^2/(16R^2)
b^2/(16R^2) = 1 - 15/16
b^2/(16R^2) = 1/16
b^2 = R^2
b = R
Substituting the values of a and b in the formula for distance traveled by Satellite 2:
2π√(16R^2 + R^2)/2
2π√(17R
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A satellite moves around the earth in a circular orbit of radius R centered at the earth. A second satellite moves in an elliptic orbit of major axis 8R, with the earth at one of the foci. If the former takes 1 day to complete a revolution, the latter would takea)21.6 daysb)8 daysc)3 hoursd)1.1 hourCorrect answer is option 'B'. Can you explain this answer?
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