Question

(●)(●)(●)(●)NEED HELP(●)(●)(●)(●)


Suppose there are 2 bodies of mass M1 and M2 sepreated by diatance "D".
Left in empty space.

( Note: Here empty space means Outer space)

And there's nothing around them within a radius of a quadrillion Lightyears.

Now these two bodies will attract each other (because of their gravitational forces). Such after some time they'll be touching each other.


Now what you have to do here is calculate that time taken by those two bodies to attract each other.

Answer 1

Hey friend, Harish here.

Here is your answer:

Given that,

1) Masses are M₁ & M₂  respectively.

2) Distance between them is d.

To find, 

Time taken for the objects to touch each other.

Solution:

We know that,

$G(M+m) = ( \frac{2 \pi }{T} )^{2}a^{3}$  

(Kepler's third law)

→ Here a is the semi major axis.  T- Time period of orbit ( From apo center to the peri center and then again continues the same motion)
 
→ Here as the move towards each other , The semi minor axis becomes zero. And does not move in elliptical orbit. 

→ The apocentre is the intial distance =  " d ". 

Then d = 2a  

→ The precenter is zero as b = 0.

Here we complete only half the orbit. Then,

$Time\ taken = \frac{Time\ period}{2}$

⇒ $\frac{T}{2} = \frac{\pi a^{3/2}}{\sqrt{G(M+m)}} = \pi\sqrt{\frac{\frac{(d)^3}{8}}{G(M+m)}} = \pi\sqrt{\frac{d^3}{8G(M+m)}}.$

$\bold{Therefore\ time\ taken\ to \ collide\ is\ \ \pi\sqrt{\frac{d^3}{8G(M+m)}}.}$
____________________________________________________

Hope my answer is helpful to you. ( Pls refer to the image for clear understanding).