Calculating time till impact

[matthewkokai]

You have 2 objects of mass (M and m). Object M is used as the frame of reference so has a velocity of 0, however object m has an initial velocity of "u". The distance between them when t=0 is d. Object m is heading straight for the centre of M.

 

Now ignoring gravity that is a simple problem (working out time till impact, t). However the reason why I don't know how to solve it is that I would like to include the effect of gravity between the objects. So I would like an equation that gives time until impact assuming that both objects are point objects and all other gravity can be ignored. I'm assuming calculus will be used.

 

Thanks.

[matthewkokai]

And the gravity of both objects is significant.

[swansont]

Use F=ma, where F = GMm/r²

 

the solution for displacement with constant acceleration is s = v₀t + 1/2 at²

 

You know that if they started at rest the two items will meet at the center of mass, so you can use the equations to find out how long it takes either one of them to get there. For the case of one object moving, you solve the equations of motion for both, simultaneously, where the impact point is now unknown, but the total distance travelled is d (so one travels a distance x, and the other a distance d-x)

[matthewkokai]

But the Force would increase as the objects get closer together. So you can't use the equations for constant acceleration.

[Meir Achuz]

You should use conservation of energy in the center of mass system.

This leads to v=\sqrt{u^2+G(m+M)/2x}, which must be integrated to find x(t).

This assumes v<<c, so SR needn't be used.

[swansont]

But the Force would increase as the objects get closer together. So you can't use the equations for constant acceleration.

 

Sorry, I left out a step: similarly, you can integrate the force equation and get s(t) for your problem.