# 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?

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Time to collision is:

# pi sqrt( (L^3 )/( 8 G(m_1+ m_2))) #

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.

From here, it's mostly just processing calculus.

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:

Not quite finished. Note that:

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The time it takes for two objects with masses ( m_1 ) and ( m_2 ) to collide in intergalactic space, assuming only gravitational forces are acting, can be calculated using the formula:

[ t = \sqrt{\frac{4l^3}{G(m_1 + m_2)}} ]

Where:

- ( t ) is the time until collision.
- ( l ) is the distance between the two objects.
- ( G ) is the gravitational constant.
- ( m_1 ) and ( m_2 ) are the masses of the two objects.

This equation is derived from the equation of motion under gravity and the conservation of energy. Once the values for ( l ), ( m_1 ), and ( m_2 ) are known, you can plug them into the formula to find the time until collision.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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