Energy conservation problem (two star system)

[Jordanosaur]

Hi guys -

Here's the problem I am chewing on:

A binary star system consists of two stars, each equal to the sun in mass. The distance between the two stars is 1.0 X 10^12m. A comet which is essentially at rest, begins to make its journey toward the binary star system as a result of gravity acting upon the comet. If the comet begins a straight line approach that will result in it passing through the midpoint of the distance between the stars, what will the velocity of the comet be at the midpoint?

I believe this to be a problem in which we need to identify the gravitational PE between the two stars, and use the answer to calculate the gravitational force acting on the comet being pulled into the star system.

I can find Ug by using the PE formula for two bodies: Ug = -G(m1m2)/r

I am coming up with roughly 2.65 X 10^8 J, but I don't understand how I can turn this around to apply to the comet's approach to the system. Any suggestions as to where I can start?

Thanks

Jordan

 

[denverdoc]

Hi guys -

Here's the problem I am chewing on:

AI believe this to be a problem in which we need to identify the gravitational PE between the two stars, and use the answer to calculate the gravitational force acting on the comet being pulled into the star system.

I can find Ug by using the PE formula for two bodies: Ug = -G(m1m2)/r

I am coming up with roughly 2.65 X 10^8 J, but I don't understand how I can turn this around to apply to the comet's approach to the system. Any suggestions as to where I can start?

Thanks

Jordan

I'm not sure that you need to worry about the PE between the stars themselves.

It may be more fruitful to consider the PE difference between infinity and the comets closest approach to one of the stars, ie the 1/2 way point between the two. Realize that you have twice this quantity from symmetry, and convert to kinetic energy. The other approach I might consider would be to realize that the center of mass of the two stars is midway between them, make an adjustment in terms of distance or mass, as two masses separated by a huge distance will in unison exert less force than two side by side.

 

[m2003]

Hi Jordan,

You are correct in identifying the gravitational potential energy (Ug) between the two stars as a key factor in this problem. However, it is important to note that the comet is also affected by the gravitational potential energy of each individual star as it approaches the binary system. This means that the total gravitational potential energy of the comet will be the sum of the individual gravitational potential energies from each star.

To calculate the gravitational force acting on the comet, you can use the equation F = -GMm/r^2, where M is the mass of one star, m is the mass of the comet, and r is the distance between the comet and the star. This will give you the net force acting on the comet due to the gravitational pull from both stars.

To determine the velocity of the comet at the midpoint between the stars, you can use the conservation of energy principle. This states that the total energy (kinetic + potential) of a system remains constant. At the midpoint, the comet will have both kinetic energy (KE) and potential energy (PE). You can set the initial potential energy (Ug) of the comet at its starting position equal to the sum of its final kinetic energy (KE) and potential energy (Ug) at the midpoint. This will give you an equation to solve for the velocity of the comet at the midpoint.

I hope this helps and gives you a starting point for solving this problem. Best of luck!

Sincerely,