Conservation of Angular Momentum and Energy

[Woodgoat]

Homework Statement

We measure a comet at perihelion to have a radial distance r1 and a velocity v1. Find the radial distance and velocity when it reaches aphelion.

Homework Equations

L=mvr=I*$$\omega$$
E=.5mv^2+U

The Attempt at a Solution

My professor skipped the chapter on Newton's Gravitation, F=GMm/r^2, so I don't think it applies here.

The book is entirely symbolic--the answer expected should be in terms of v1 and r1.

I labeled the radial distance and velocity r2 and v2, respectively. Since L is constant, m*v1*r1 = m*v2*r2, or v1*r1=v2*r2. Of course, this equation only gives the unknown radial distance in terms of the unknown velocity, or vice versa.

I don't think U = -GMm/r would apply here, since, again, my professor skipped the chapter on gravitation. I tried solving using a second equality, .5m*v1^2-GMm/r1=.5m*v2^2-GMm/r2, but when I solved the system of equations with Mathematica, I got a gigantic mess for an answer.

I appreciate your time.

 

[Google_Spider]

I think both your equations correct, you just did something wrong with the calculations.

BTW, what's mathematica ? Are you using a http://en.wikipedia.org/wiki/Mathematica" to solve these equations ?

 

[Moetasim]

Conservation of angular momentum and energy are fundamental laws of physics that apply to many different systems, including the motion of comets. In this situation, we can use these principles to find the radial distance and velocity of the comet when it reaches aphelion.

Firstly, let's consider the conservation of angular momentum, which states that the angular momentum of a system remains constant unless an external torque is acting on it. In this case, the only external force acting on the comet is the gravitational force, which is central and therefore does not produce any torque. This means that the angular momentum of the comet remains constant throughout its motion.

We can express the angular momentum of the comet as L = mvr, where m is its mass, v is its velocity, and r is the radial distance from the center of the orbit. Since the angular momentum is constant, we can equate the values at perihelion (r1 and v1) to the values at aphelion (r2 and v2). This gives us the equation mvr1 = mvr2, or v1r1 = v2r2.

Next, let's consider the conservation of energy, which states that the total energy of a system remains constant. We can express the total energy of the comet as the sum of its kinetic energy (½mv^2) and its potential energy (U = -GMm/r). At perihelion, the comet has the maximum kinetic energy and minimum potential energy, and at aphelion, it has the minimum kinetic energy and maximum potential energy. Therefore, we can equate the values at perihelion to the values at aphelion, giving us the equation ½mv1^2 - GMm/r1 = ½mv2^2 - GMm/r2.

Now, we have two equations (v1r1 = v2r2 and ½mv1^2 - GMm/r1 = ½mv2^2 - GMm/r2) with two unknowns (r2 and v2). We can solve this system of equations to find the values of r2 and v2. This can be done algebraically or by using a software program like Mathematica.

In conclusion, by applying the principles of conservation of angular momentum and energy, we can find the radial distance and velocity of the comet at aphelion. These principles are important tools in understanding the motion of celestial