The distance between two objects with mass m1 and m2 are in the intergalactic space is equal to l. With no other forces acting except gravity, how soon will they collide?
2 Answers
Assuming that object of mass
We make use of Kepler's Third Law which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis
#T^2propa^3#
It is given that the distance between the two objects is
- Using the concept of reduced mass
#bbmu# which is a quantity which allows the two-body problem to be solved as if it were a one-body problem. We get#bbmu=1/(1/m_1+1/m_2)=(m_1m_2)/(m_1+m_2)# ....(1)
where the force on this mass is given by the force between the two objects. - For a circular orbit of radius
#r# equating centripetal force with the gravitational force we have#bbmuromega^2=G(m_1m_2)/r^2# .....(2)
Rewriting
#(m_1m_2)/(m_1+m_2)l/2((2pi)/T)^2=G(m_1m_2)/(l/2)^2#
#=>(1)/(m_1+m_2)(l/2)^3(4pi)/T^2=G#
#=>T=pisqrt(l^3/(2G(m_1+m_2)))#
3. In the limiting case, the two objects are moving in a straight line towards each other. If#T_c# is the time of collision, then
#T_c=T/2=pi/2sqrt(l^3/(2G(m_1+m_2)))#
Time to collision is:
# pi sqrt( (L^3 )/( 8 G(m_1+ m_2))) #
Explanation:
KEY IDEA : Because there are no external forces on the system, the centre of mass (CoM) of the system will remain the same throughout, and this is also where any collision will occur.
Makes sense to place the origin of a co-ordinate system at the CoM with the objects on a number line such that, at
#implies {(m_1: qquad x(0) = x_o = - m_2/m_1 y_o),(m_2: qquad y(0) = y_o):}#
Throughout, that same relationship,
Considering only
From here, it's mostly just processing calculus.
Using integrating factor,
IV's :
Bit of physics : Take the negative square root as
That is separable . And if the objects collide at
In terms of the indefinite version of the integral in red, this intermediate step is all over the internet, and it works really well:
That intermediate step sets it up very nicely for IBP. Noting that:
Then by IBP :
ie:
The stated initial separation of the objects,
#L = y - x = y_o(1 + m_2/m_1)#
Not quite finished. Note that: