Newton's Law of Universal Gravitation (Differential Equation Question)

In summary, the conversation involves a student seeking help with a problem involving a differential equation and a definite integration. The conversation includes discussions on how to handle constants and limits, and how to find the required answer.
  • #1
Bazzinga
45
0
I'm having trouble with part a) of this question...

[PLAIN]http://img69.imageshack.us/img69/5815/98157006.png


So I started off by solving the DE above a), and I've gotten it down to:

[tex]\frac{1}{2} m v^{2} = \frac{mgR^{2}}{(x + R)} + C[/tex]

I can tell I'm getting close, but I'm a little confused where h is coming from, and how to get rid of the [tex]R^{2}[/tex]. Also, what do I do with C??

I know it says x = x(t) which is the height, so does that mean that x(t) = h? I've never taken physics (and this was in my Calculus homework) so this is all new to me!
 
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  • #2
Good so far, to find C set v=0 when x=h, then to find v_0 set x=0.
 
  • #3
Hi, you should use a definite integration

[tex] \int_{v_0}^{0} mv \ dv = \int_{0}^{h} -\frac{mgR^2}{(R+x)^2} dx [/tex]

That way you'll get the formula in the book.

As for the limit, I think it's trivial, right ?
 
  • #4
or, he could have done what I told him to to find the constant C, both methods are equivalent.
 
  • #5
So C = (-mgR^2) / (h+R) ?
 
  • #6
Correct. From there you can easily derive the required answer.
 
  • #7
Thanks a lot guys! So what about b)? will V0 approach infinity because the top approaches infinity faster than the bottom?
 
  • #8
divide the top and bottom of the fraction and you will have on the demoninator a R/h which is the only place h appears, take the limit as h tends to infinity and that yerm dissapears.
 

Related to Newton's Law of Universal Gravitation (Differential Equation Question)

1. What is Newton's Law of Universal Gravitation?

Newton's Law of Universal Gravitation is a scientific law that describes the force of gravity between two objects. It states that the force of gravity is directly proportional to the masses of the objects and inversely proportional to the square of the distance between them.

2. How is Newton's Law of Universal Gravitation expressed mathematically?

Newton's Law of Universal Gravitation can be expressed as F = G (m1m2/r^2), where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

3. What is the significance of the gravitational constant in Newton's Law of Universal Gravitation?

The gravitational constant, denoted by G, is a proportionality constant that relates the force of gravity to the masses and distance between objects. It allows us to calculate the strength of the gravitational force between any two objects in the universe.

4. What is the role of differential equations in Newton's Law of Universal Gravitation?

Differential equations are used to mathematically model the motion of objects under the influence of gravity. By solving the differential equations derived from Newton's Law of Universal Gravitation, we can predict the trajectory and behavior of objects in space.

5. Can Newton's Law of Universal Gravitation be applied to all objects in the universe?

Yes, Newton's Law of Universal Gravitation applies to all objects with mass in the universe, from small objects on Earth to large celestial bodies. However, it may not accurately describe the behavior of objects at very small scales, such as subatomic particles.

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