A small particle is dropped from a height `R` in front of a narow tunnel dug inside the earth (along a diameter). Let `M` be the mass of earth, `R` be radius of earrth. Let `v_(0), T` be speed of particle when it reaches `A` and time taken by particle to go from `A` to `B` respectively. Assuming mass of particle to be negligible as compare to mass of earth, pick the correct option(s)

Options

A. `V_(0)=sqrt((GM)/(2R))`
B. `V_(0)=sqrt((GM)/R)`
C. `R=(pi)/2sqrt((R^(3))/(GM))`
D. `T=pisqrt((R^(3))/(GM))`

Answer

B, C

Explanation

`(-GMm)/(2R)=(-GMm)/R+1/2mv_(0)^(2)`
`impliesv_(0) =sqrt((GM)/R)`
while in the tunnel,
`(d^(2)x)/(dt^(2))=-((GM)/(R^(3)))x`
`v_(0)^(2)=(GM)/R=omega^(2)(a^(2)-R^(2))impliesa=sqrt(2)R`
`implies T_(AtoB)=1/4xx2pisqrt((R^(3))/(GM))`