In free space, two particles of mass m each are initially both at rest at a distance a from each other. They start moving towards each other due to their mutual gravitational attraction. The time after which the distance between them has reduced to `(a)/(2)` is:

Options

A. `((pi+2)/(4sqrt(2)))((a^(3))/(Gm))^(1//2)`
B. `((pi-2)/(4sqrt(2)))((a^(3))/(Gm))^(1//2)`
C. `((pi+2)/(8))((a^(3))/(Gm))^(1//2)`
D. `((pi-2)/(8))((a^(3))/(Gm))^(1//2)`

Answer

C

Explanation

Let the origin be at the CM of the particles, let the initial positions of the particles be `x=(a)/(2)` and `x=-(a)/(2)` and let the instantaneous positions of the particles be x = r and x = -r
Let the instantaneous velocity of each particle be v
Let the time after which the distance between the particles has reduced to `(a)/(2)` be T
Then, for the particle that was initially at `x=(a)/(2)`,
`(Gm^(2))/((2r)^(2))=-m("dvd")/(dr)" "implies" "(Gm)/(r^(2))=-(4vdv)/(dr)`
`implies" "-4int_(0)^(v)dvd=Gm int_(a//2)^(r)(dr)/(r^(2))impliesv^(2)=(Gm)/(2)((1)/(r)-(2)/(a))impliesv=-[(Gm)/(2)((1)/(r)-(2)/(a))]^(1//2)`
[v is negative because the velocity is towards the -X direction]
`implies" "(dr)/(dt)=-[(Gm)/(2)((1)/(r)-(2)/(a))]^(1//2)implies-int_(a//2)^(a//4)sqrt((r)/(a-2r))dr=sqrt((Gm)/(2a))int_(0)^(T)dt`
`implies" "int_(a//2)^(a//4)sqrt((r)/(a-2r))dr=-sqrt((Gm)/(2a))T" "...(i)`
Let us now evaluate the integral `I=int sqrt((r)/(a-2r))dr`
Let `r=(a sin^(2)theta)/(2)" "implies" "dr=a sin theta cos theta d theta`
Therefore, `I=(a)/(sqrt(2))int sin^(2)theta d theta=(a)/(2sqrt(2))int(1-cos2theta)d theta=(a)/(2sqrt(2))[theta-(1)/(2)sin 2theta]`
Since `r=(a sin^(2)theta)/(2),theta=sin^(-1)sqrt((2r)/(a))` and `sin2theta=2sin theta cos ttheta=2(sqrt((2r)/(a)))(sqrt(1-(2r)/(a)))=(sqrt(8r(a-2r)))/(a)`
So, `I=(a)/(2sqrt(2))(sin^(-1)sqrt((2r)/(a))-(sqrt(2r(a-2r)))/(a))=(a)/(2sqrt(2))sin^(-1)sqrt((2r)/(a))-(sqrt(r(a-2r)))/(2)`
Therefore, from equation (i),
sin `T=-sqrt((2a)/(Gm))((a)/(2sqrt(2))sin^(-1)sqrt((2r)/(a))-(sqrt(r(a-2r)))/(2))_(a//2)^(a//4)=-sqrt((2a)/(Gm))(-(a)/(2sqrt(2))((pi)/(4))-(1)/(2)((a)/(2sqrt(2))))`
Hence, `" "T=sqrt((a)/(Gm))((api)/(8)+(a)/(4))=((pi+2)/(8))sqrt((a^(3))/(Gm))`