Calculating Time to Collision: Using Integration and Gravity

[nealh149]

This is not a homework question, but came about because of a homework question. I am now curious.

How would you go about (using calculus) finding the time it will take for two objects attracted by gravity to collide. I know it will involve integrating the acceleration, but I cannot come up with a formula that will define gravitational acceleration with respect to time.

Any help would be great.

 

[arunma]

That's simple. Solve the differential equation,

$$\frac{d^2 r}{dt^2}=\frac{GM}{r^2}$$

Here r is the position of one object with respect to the second's center of mass, and M is the mass of the second object. Just figure out how close the objects' centers of mass need to be in order for them to collide (for a sphere, this would be when r equals the sphere's radius). If both objects are of comparable size, you'll also need to consider the fact that both objects are accelerating.

Now as to how you'd solve that differential equation, I really have no clue (Laplace Transform, maybe?). But I'm a physics person, so I'd be lazy, plug it into Mathematica, and see what I get.

 

[Thrawn]

That's simple. Solve the differential equation,

$$\frac{d^2 r}{dt^2}=\frac{GM}{r^2}$$

Here r is the position of one object with respect to the second's center of mass, and M is the mass of the second object. Just figure out how close the objects' centers of mass need to be in order for them to collide (for a sphere, this would be when r equals the sphere's radius). If both objects are of comparable size, you'll also need to consider the fact that both objects are accelerating.

Now as to how you'd solve that differential equation, I really have no clue (Laplace Transform, maybe?). But I'm a physics person, so I'd be lazy, plug it into Mathematica, and see what I get.

What does the bottom part of the left side mean (dt)?

EDIT: And for anyone who DOES know, How WOULD one go about solving this?

 

[arunma]

What does the bottom part of the left side mean (dt)?

It's a second derivative of position with respect to time.

EDIT: And for anyone who DOES know, How WOULD one go about solving this?

In general, second order non-linear differential equations aren't very easy to solve. But here's the methods I'd probably try (in order of effectiveness) if I wanted an analytical solution.

-Guess
-Apply the Laplace Transform
-Look for a series solution
-Plot the numerical solution, and see if it looks like any obvious explicit function

Of course, it's distinctly possible that the equation has no analytical solution. Most differential equations don't.

 

[Thrawn]

It's a second derivative of position with respect to time.



In general, second order non-linear differential equations aren't very easy to solve. But here's the methods I'd probably try (in order of effectiveness) if I wanted an analytical solution.

-Guess
-Apply the Laplace Transform
-Look for a series solution
-Plot the numerical solution, and see if it looks like any obvious explicit function

Of course, it's distinctly possible that the equation has no analytical solution. Most differential equations don't.

And what would the Laplace Transform be? Does it help if I know the end velocity of the object?

 

[hage567]

Just curious: how can you know the end velocity if you don't know the acceleration??

Have you looked at the following link to see what Laplace transforms are all about?

http://en.wikipedia.org/wiki/Laplace_transforms

I'm not trying to discourage you, but these aren't a trivial matter. This is upper level university math, and you're not even familiar with calculus yet!
I'm not sure what advice to give you about this. What is this work for exactly, a science fair or something?

 

[Thrawn]

Just curious: how can you know the end velocity if you don't know the acceleration??

Have you looked at the following link to see what Laplace transforms are all about?

http://en.wikipedia.org/wiki/Laplace_transforms

I'm not trying to discourage you, but these aren't a trivial matter. This is upper level university math, and you're not even familiar with calculus yet!
I'm not sure what advice to give you about this. What is this work for exactly, a science fair or something?

The end velocity should be equivalent to the escape velocity of the attractive mass, so I found that.

On another note, does anyone know of a website (or book) which gives an introduction to calculus, starting at the very basics then working up?

 

[hage567]

Have you checked the PF tutorials section? There are links there to lots of resources. You could just go to a library to see if there are any books on the subject. You can pick one out that you like by doing it that way.

 

[amacleod]

I hope it's ok for me to jump in on a thread that's a few months old--I ran across this forum when googling for differential equations.

Just curious: how can you know the end velocity if you don't know the acceleration??

Energy. The increase in kinetic energy is equal to the decrease in potential energy. At the time of collision, all the gravitational potential energy between the two bodies will have been converted to kinetic energy.

$$\Delta KE = \frac{1}{2}mv^{2}$$

Potential energy due to gravitation is

$$U_{G} = GMm\left(\frac{1}{h_{1}} - \frac {1}{h_{2}}\right)$$

where $$h_{1}$$ is the distance between centers of mass at time of collision (essentially the sum of the radii, if the bodies are spheres) and $$h_{2}$$ is the starting distance between centers of mass. So, simply using energy, it's possible to solve for change in velocity.

$$v = \sqrt{2GM\left(\frac{1}{h_{1}} - \frac {1}{h_{2}}\right)}$$

Unfortunately, this doesn't help to find either acceleration or time.

$$\Delta v = a_{avg}t$$

If we knew one of average acceleration or time, we could find the other, but it looks like the differential equation is necessary to find either of them. It may be some small consolation that the average acceleration should at least be between the initial acceleration and the final acceleration, since within the constraints of the problem, acceleration will be a continuous monotonically increasing function of time.

$$a_{0} < a_{avg} < a_{final}$$

 

[Irid]

I suggest you consider the position of mass center of the system. Use this to eliminate one unknown, and you'll be able to integrate the equation in no time.