Energy conservation + planets

In summary, it is important to consider both the energy and momentum conservation equations when calculating the speed of masses in a gravitational system. Additionally, the approximation of one mass being much larger than the other may not always be accurate or helpful.
  • #1
NAkid
70
0

Homework Statement


Two particles of masses M1 = 81000 kg and M2 = 9400 kg are initially at rest an infinite distance apart. As a result of the gravitational force the two masses move towards each other. Calculate the speed of mass M1 and mass M2 when their separation distance is 26.5 km.


Homework Equations





The Attempt at a Solution


I think you have to use energy-conservation equations here for each mass

1/2mv1^2 - (GM1M2)/R = 1/2mv2^2 - (GM1M2)/R

so, for mass M1, v1=0 and (GM1M2)/R on the left = 0 because R = infinite. then you get

1/2m1v2^2 = (GM1M2)/R --and solve for v2, but this doesn't seem to be correct..
 
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  • #2
Lets say that the 2 masses were infinitely separated. The potential energy of the system would be 0J. However, an attractive force acts on them and they are brought closer together, in which the separation between them 26.5 km. Because it is an attractive force, the potential energy of the system is negative. What you do is find the change in potential energy. The initial value is 0J and the final value would be when they are the given distance apart. Using -dU = dK (d being delta, U being gravitational potential energy, and K being kinetic energy), you would solve for the kinetic energy of each mass. From that, you solve for velocity.
 
  • #3
isn't that basically what i did? calculated the changes to solve for velocity?
 
  • #4
Hint:

M1>>M2
 
  • #5
physixguru said:
Hint:

M1>>M2

M1 / M2 <10 so I don't think you can say that.


-GM1M2/r is the potential energy of BOTH the masses.

At the start, potential and kinetic energy is 0. at a distance r the total energy is:

[tex](1/2)m_1v_1^2 + (1/2)m_2v_2^2 - Gm_1m_2/r[/tex]

This won't allow you to calculate [tex]v_1[/tex] and [tex]v_2[/tex] by itself. You also need conservation of momentum.
 
  • #6
kamerling said:
M1 / M2 <10 so I don't think you can say that.


-GM1M2/r is the potential energy of BOTH the masses.

At the start, potential and kinetic energy is 0. at a distance r the total energy is:

[tex](1/2)m_1v_1^2 + (1/2)m_2v_2^2 - Gm_1m_2/r[/tex]

This won't allow you to calculate [tex]v_1[/tex] and [tex]v_2[/tex] by itself. You also need conservation of momentum.

I never told him to ignore the masses.

DID I?
 
  • #7
physixguru said:
I never told him to ignore the masses.

DID I?

Well I didn't think you said that. I just don't think that M1>>M2 is either true or helpful.
 
  • #8
kamerling said:
Well I didn't think you said that. I just don't think that M1>>M2 is either true or helpful.

It is helpful for sure, especially when energy consideration comes into play.
 
  • #9
physixguru said:
It is helpful for sure, especially when energy consideration comes into play.
It is not helpful for this problem. First, the approximation m1>>m2 is incorrect here, as m2 > m1/10. Second, the approximation implicitly makes the velocity of the larger mass zero, and the problem asks for both v1 and v2.

kamerling said:
-GM1M2/r is the potential energy of BOTH the masses.

At the start, potential and kinetic energy is 0. at a distance r the total energy is:

[tex](1/2)m_1v_1^2 + (1/2)m_2v_2^2 - Gm_1m_2/r[/tex]

This won't allow you to calculate [tex]v_1[/tex] and [tex]v_2[/tex] by itself. You also need conservation of momentum.

This, on the other hand, is very good advice.
 

1. How does energy conservation affect the planets?

Energy conservation plays a crucial role in protecting and preserving the planets. By reducing our energy consumption and using more sustainable sources, we can minimize the negative impact on the environment and help mitigate climate change.

2. What are some ways to conserve energy on a planetary scale?

Some effective ways to conserve energy on a planetary scale include using renewable energy sources, improving energy efficiency in buildings and transportation, and promoting sustainable practices such as recycling and reducing waste.

3. How does energy conservation impact the future of our planets?

Energy conservation is essential for the long-term sustainability of our planets. By reducing our reliance on fossil fuels and transitioning to cleaner energy sources, we can help ensure a healthier and more habitable planet for future generations.

4. Can energy conservation help mitigate the effects of climate change on the planets?

Yes, energy conservation can play a significant role in mitigating the effects of climate change on the planets. By reducing greenhouse gas emissions, we can slow down the rate of global warming and protect the planets from extreme weather events and other consequences of climate change.

5. What can individuals do to contribute to energy conservation on the planets?

Individuals can make a significant impact on energy conservation by making small changes in their daily habits, such as turning off lights and electronics when not in use, using public transportation or carpooling, and supporting renewable energy initiatives. Advocating for sustainable policies and practices can also make a difference on a larger scale.

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