Integration and Gravity

In summary: It can be converted to other forms, like work. In summary, the end velocity should be equivalent to the escape velocity of the attractive mass.
  • #1
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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.
 
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  • #2
That's simple. Solve the differential equation,

[tex]\frac{d^2 r}{dt^2}=\frac{GM}{r^2}[/tex]

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.
 
  • #3
arunma said:
That's simple. Solve the differential equation,

[tex]\frac{d^2 r}{dt^2}=\frac{GM}{r^2}[/tex]

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?
 
  • #4
Thrawn said:
What does the bottom part of the left side mean (dt)?

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

Thrawn said:
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.
 
  • #5
arunma said:
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?
 
  • #6
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?
 
  • #7
hage567 said:
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?
 
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  • #8
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.
 
  • #9
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.

hage567 said:
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.

[tex]\Delta KE = \frac{1}{2}mv^{2}[/tex]

Potential energy due to gravitation is

[tex]U_{G} = GMm\left(\frac{1}{h_{1}} - \frac {1}{h_{2}}\right)[/tex]

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

[tex]v = \sqrt{2GM\left(\frac{1}{h_{1}} - \frac {1}{h_{2}}\right)}[/tex]

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

[tex]\Delta v = a_{avg}t[/tex]

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.

[tex]a_{0} < a_{avg} < a_{final}[/tex]
 
  • #10
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.
 

1. What is integration in the context of gravity?

Integration in the context of gravity refers to the process of combining multiple gravitational forces to determine the overall effect on an object. This is done using mathematical integration techniques to calculate the total force and direction of the gravitational pull.

2. How does integration relate to Newton's Law of Universal Gravitation?

Integration is a key tool used to understand and apply Newton's Law of Universal Gravitation, which states that every object in the universe exerts a force of attraction on every other object. By integrating the individual gravitational forces between two objects, we can determine the overall force between them.

3. What are some examples of integration in the study of gravity?

Examples of integration in the study of gravity include calculating the gravitational force between planets in our solar system, determining the trajectory of a spacecraft or satellite, and predicting the motion of objects in orbit around a central body.

4. How does integration play a role in understanding the concept of gravity as a curved spacetime?

Integration is a crucial tool in understanding gravity as a curved spacetime in Einstein's theory of general relativity. By integrating the curvature of spacetime caused by massive objects, we can predict how objects will move in this curved space, which is what we experience as the force of gravity.

5. Can integration be used to explain other phenomena related to gravity?

Yes, integration can be used to explain other phenomena related to gravity, such as gravitational waves and the bending of light by massive objects. These effects can be understood by integrating the changes in gravitational forces as they propagate through space or interact with other objects.

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