I Time for two objects to collide due to gravity

Tags:
1. May 15, 2016

LightningBolt226

When two objects, there is a change in the force they exert on each other due to gravity because of the change in distance between them. Therefore, there's also changing acceleration. How do I find the time it takes for the two to collide? Their final velocities?
I've already looked at quite a few other threads on this forum but I still haven't been able to find one that answers this question nor gives me any clues on how to proceed.
These 2 links proved to be particularly helpful. There's just a part I don't quite understand because of arildno's answer on the second thread which has messed up LaTeX formatting which was referred to in rcqldr's answer on the first thread.
I don't quite understand where u came from nor what it represents.

This is what I've come up with so far on my own:

I originally tackled this as an object approaching a planet so I only looked at it from the perspective of the object.

Finding the time taken:
$$v-v_{0} = a_{0}t + \frac{1}{2}jt^{2}$$
$$\frac{1}{2}jt^{2}+a_{0}t-(v-v_{0})=0$$
$$t=\frac{-a_{0}\pm \sqrt{a_{0}^{2}+2j(v-v_{0})}}{j}$$

Finding acceleration:
$$a=\frac{GM}{r^{2}}$$

Finding jerk. I read on another thread that the derivative of gravitational force is jerk. Is this right? What's making me wonder is the fact that it's negative. Does that mean the acceleration is decreasing?
$$j=\frac{\partial F}{\partial r}=\frac{\partial }{\partial r}\left(\frac{GmM}{r^{2}}\right)=-\frac{2GMm}{r^{3}}$$

Finding final velocity:
$$KE_{i} + PE_{i} = KE_{f} + PE_{f}$$
$$\frac{1}{2}mv_{i}^{2}-\frac{GMm}{r_{i}}=\frac{1}{2}mv_{f}^{2}-\frac{GMm}{r_{f}}$$
Assuming that the other object has no initial velocity, the starting KE can be cancelled out.
$$\frac{1}{2}v_{f}^{2}=\frac{GM}{r_{f}}-\frac{GM}{r_{i}}$$
$$v=\sqrt{\frac{2GM(r_{i}-r_{f})}{r_{i}r_{f}}}$$

2. May 15, 2016

Staff: Mentor

All the quantities (position, velocity, acceleration, jerk, ...) will vary in a nonlinear way. You cannot use equations that require them to be linear, you'll need a differential equation to solve.
Depends on your sign convention, if acceleration is negative then its increase in magnitude gives a negative jerk.

3. May 15, 2016

m4r35n357

This is actually not a trivial problem. I believe that the maths you need is here. This first example is worked for Newtonian gravity, but as it happens the solution translates identically to GR if you use proper time as the time variable (this is how we can work out the proper time to "hit the singularity").

4. May 16, 2016

Khashishi

This is a classical central force problem.
(cf https://en.wikipedia.org/wiki/Classical_central-force_problem)
A two body problem can easily be turned into a one-body problem using the reduced mass.
If they are traveling in a straight line toward each other, it's simpler still.
You can use the equation given in the wiki, changing m to mu, and taking r to mean the relative distance.
$\mu = \frac{mM}{m+M}$
$|t-t_0| = \sqrt{\frac{\mu}{2}} \int \frac{|dr|}{\sqrt{E_\text{tot} - U(r)}}$
$U(r) = -\frac{mMG}{r}$
$E_\text{tot}=U(r_0) + \frac{1}{2}\mu v_0^2$
Try to take it from there

5. May 16, 2016

rcgldr

u was used for a substitution to help solve that last integral. There is a more recent thread, using an alternate substitution (θ) for the last integral.

You'll need to need to determine θ0 and θ1 based on r. If these are point masses, then

$θ_0 = \frac{\pi}{2}$
$θ_1 = 0$

Last edited: May 16, 2016
6. May 18, 2016

LightningBolt226

Thanks so much for the link. I understand the solution now. I just have one question. There was a part where θ was defined as
$$r=r_{0}sin^{2}θ$$

7. May 18, 2016

rcgldr

It's not an identity, it's a definition of θ based on observation designed to simplify the integral, so that after substitution, the square root expression becomes tan(θ).

8. May 18, 2016

LightningBolt226

Sorry, I meant the definition. I don't understand how the equation was formed. Could you please elaborate? Thanks :)

9. May 18, 2016

rcgldr

Either by observation or trial and error. Starting off with r / (r0 - r), you choose some substitution r = r0 f(θ), in which case r / (r0 - r) becomes r0 f(θ) / (r0 - r0 f(θ)) = r0 (f(θ) / (1 - f(θ))), and at this point some insight leads to choosing f(θ) = sin2(θ). (f(θ) = cos2(θ) could have also been chosen.)

10. May 18, 2016

LightningBolt226

Oh. I get it now. Thanks so much! :)