An Interesting Gravitation Problem

In summary, the minimum distance between the second satellite and the planet is when the potential energy is the least.
  • #1
k1point618
25
0

Homework Statement


Two satellites are launched at a distance R from a planet of negligible radius. (Yes, that's what the problem says...) Both satellites are launched in the tangential direction. the first satellite launches correctly at a speed [tex]v_0[/tex] and enters a circular orbit. The second satellite, however, is launched at a speed [tex]\frac{1}{2}v_0[/tex] . What is the minimum distance between the second satellite and the planet over the course of its orbit?


Homework Equations





The Attempt at a Solution


I thought about using energy. The two satellites both start out with the same potential energy but different kinetic energy.

So satellite one's TME: [tex]-\frac{GMm}{R} + \frac{1}{2}mv_0^2[/tex]

where as satellite two's TME:[tex]-\frac{GMm}{R} + \frac{1}{8}mv_0^2[/tex]

And the second satellite's minimum distance is when its potential is the least...
Somehow I think this problem might relate to angular momentum... L = mvr, but not exactly sure.

THANK YOU =D
 
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  • #2
You can use the fact the first satellite goes into a circular orbit to figure out M in terms of R and v0 using the centripetal acceleration of an object in circular motion. Let R1 and v1 be the radius at the lowest point. At the lowest point v1 is again tangenential to the planet. So conservation of angular momentum gives you one equation in R1 and v1 and conservation of energy gives you another. Solve them simultaneously.
 
  • #3
Ah, i c~

and i got it right =D

ThanX
 
  • #4
Dick said:
You can use the fact the first satellite goes into a circular orbit to figure out M in terms of R and v0 using the centripetal acceleration of an object in circular motion. Let R1 and v1 be the radius at the lowest point. At the lowest point v1 is again tangenential to the planet. So conservation of angular momentum gives you one equation in R1 and v1 and conservation of energy gives you another. Solve them simultaneously.

How can you prove that the lowest point does indeed occur when the velocity is tangent again? And why do we need to know that it is tangent? Is something not conserved otherwise?
 
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  • #5
compwiz3000 said:
How can you prove that the lowest point does indeed occur when the velocity is tangent again? And why do we need to know that it is tangent? Is something not conserved otherwise?

The lowest point occurs when the object is traveling 'horizontally' i.e. perpendicular to the radial vector connecting the object to the planets center. Draw a picture. Knowing the angle between those vectors makes it easy to compute angular momentum. Try it.
 
  • #6
haha I forgot that angular momentum is a cross product
 

Related to An Interesting Gravitation Problem

1. What is the problem being discussed in "An Interesting Gravitation Problem"?

The problem being discussed is a hypothetical situation where two objects with different masses and velocities are placed in a gravitational field and asked to determine the distance between them at a certain time.

2. What is the significance of this problem in the field of physics?

This problem helps to understand the concept of gravitational force and its effects on objects with different masses and velocities. It also allows for the application of mathematical equations and principles to analyze and solve the problem.

3. How is this problem solved?

This problem can be solved using the equations of motion and Newton's law of universal gravitation. The equations can be manipulated and solved simultaneously to determine the distance between the objects at a specific time.

4. Are there any assumptions made in solving this problem?

Yes, there are a few assumptions made in solving this problem. The objects are assumed to be point masses, the gravitational force is assumed to be constant, and there are no external forces acting on the objects.

5. What are some real-life applications of this problem?

This problem has many real-life applications, such as calculating the trajectory of satellites in orbit, predicting the motion of planets in the solar system, and understanding the behavior of objects in gravitational fields.

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