An artificial satellite (mass m) of a planet (mass M) revolves in a circular orbit whose radius is n times the radius R of the planet in the process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming the force of resistance on satellite to depend on velocity as `F=av^(2)` where 'a' is a constant caculate how long the satellite will stay in the space before it falls on to the planet's surface.

Options

A. `(m sqrt(R) ( sqrt(n) - 1))/(a sqrt(GM))`
B. `(m sqrt(R) ( sqrt(n) + 1))/(a sqrt(GM))`
C. `(2m sqrt(R) ( sqrt(n) - 1))/(a sqrt(GM))`
D. `(m sqrt(R) ( sqrt(n) - 1))/(a sqrt(2GM))`

Answer

A

Explanation

F = `(mdv)/(dt)= av^(2)`
`sqrt((GM)/(R))`
`m int_(sqrt((GM)/(nR)))^(R) (dv)/(v^(2)) = a int_(0)^(t) dt `
`([ - sqrt((R)/(GM)) + sqrt((nR)/(GM)) ] m)/(a) = t`
`t = (m sqrt(R) (sqrt(n) - 1))/(a sqrt(GM))`
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