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A point mass is orbiting a significant m...

A point mass is orbiting a significant mass `M` lying at the focus of the elleptical orbit having major and minor axes given by `2a` and `2b` respectively. Let `r` be the distance between the mass `M` and the point of major axis. The velocity of the particle can be given as

A

`(ab)/(2r) sqrt((GM)/(a^(3)))`

B

`(ab)/rsqrt((GM)/(b^(3)))`

C

`(ab)/rsqrt((GM)/(a^(3)))`

D

`(2ab)/rsqrt((GM)/(((a+b)/2)^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
A
on stopping, the satellite will fall along the radius `r` of the orbit which can be regarded as a limiting case of an ellipse with semi-major axis `r/2` Using kepler's third law `T^(2)propr^(3)`
time of fall `(T')/2=T/(2sqrt(8))=(sqrt(2)T)/8`
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Knowledge Check

  • A point mass is orbiting a significant mass M lying at the focus of the elliptical orbit having major and minor axes given by 2 a and 2b respectively. Let r be the distance between the mas M and the end point of major or axis. Velocity of the particle can be given as

    A
    `(ab)/r sqrt((GM)/(a^(3)))`
    B
    `(ab)/rsqrt((GM)/(b^(3)))`
    C
    `(ab)/(2r)sqrt((GM)/(r^(3)))`
    D
    `(2ab)/rsqrt((GM)/(((a+b)/2)^(3)))`

  • When earth is at one end of the major axis of the elliptical orbit having major and minor axes 2A and 2B , respectively, its velocity (with magnitude V_0 ) makes an angle theta with the major axis. What is the value of theta and what may be the areal velocity of the earth ?

    A
    `0^@ and 0.5 V_0 [A+sqrt(A^2-B^2)]`
    B
    `0^@ and 0.5 V_0 [A-sqrt(A^2-B^2)]`
    C
    `90^@ and 0.5 V_0 [A+sqrt(A^2-B^2)]`
    D
    `90^@ and 0.5 V_0 [A-sqrt(A^2+B^2)]`

  • A satellite of mass m orbits around the Earth of mas M in an elliptical orbit of semi - major and semi - minor axes 2a and a respectively. The angular momentum of the satellite about the centre of the Earth is

    A
    `pimsqrt((GMa)/(4))`
    B
    `pim sqrt((GMa)/(4))`
    C
    `msqrt((GM a)/(8))`
    D
    `msqrt((GM a)/(2))`

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