Intermediate Mechanics Practice Test Problem

In summary, this is a problem involving an isolated two-body system with a coordinate transformation to a non-inertial frame. The conservation laws of angular momentum and mechanical energy are applied to find the necessary radial velocity, which is found to be 9 times the square root of the product of the masses and the gravitational constant divided by the initial radius. The trajectory is a hyperbola in the frame of reference of one of the masses.
  • #1
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Hello everyone. The following is from the practice test given to me for my intermediate mechanics final. I'm, at the moment, completely lost on what to do. If you have even just a few ideas of what I should be doing they would be apperciated. If you want to give a detailed solution it would be wonderful.

Problem: The relative coordinate of a system of two masses, m and M, is moving in a circular orbit of radius r_0 because of the mutual gravitational attraction. Suddenly the particle with mass m receives a kick in negative radial direction. As a consequence of this kick, the particles approach each other until the centrifugal force finally repels them again.

What radial velocity, v_0 as a result of the kick is necessary to have the particles approach each other to a minimum distance of 10% of r_0?
Is the new trajectory a circle, ellipse, parabola or hyperbola?

Thank you for your time.
 
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  • #2
Hello, this is an interesting question. The following is my opinion only and something detail won't be metioned.

The system in the question is an isolated two-body one.
One can apply a coordinate transformation from the usual inertial frame to the relative-motion frame, i.e. observe m on M.
Acturally, the frame is a non-inertial one.
But one can make correction: [tex]m\rightarrow\frac{Mm}{M+m}[/tex], which let Newton's law applied well.
Therefore, on the [tex]M[/tex]-frame, after [tex]V_0[/tex] being added on [tex]m[/tex], the conservation law of angular momentum and mechanical energy should be satisfied.
From the beginning point (relative radius [tex]r_0[/tex] , tangent velocity [tex]\sqrt{\frac{G(M+m)}{r_0}}[/tex], radial velocity [tex]V_0[/tex]) to the nearest point (relative radius [tex]\frac{r_0}{10}[/tex], (assume) tangent velocity [tex]U[/tex] , no radial velocity):
Angular momentum conservation:
[tex]\frac{Mm}{M+m}\left(\sqrt{\frac{G(M+m)}{r_0}}\right)r_0=\frac{Mm}{M+m}U\frac{r_0}{10}\Rightarrow U=10\sqrt{\frac{G(M+m)}{r_0}}[/tex]
Mechanical energy conservation:
[tex]-\frac{GMm}{r_0}+\frac{1}{2}\frac{Mm}{M+m}\left(\sqrt{\frac{G(M+m)}{r_0}}^2+V_0^2\right)=-\frac{GMm}{r_0/10}+\frac{1}{2}\frac{Mm}{M+m}U^2\Rightarrow V_0=9\sqrt{\frac{G(M+m)}{r_0}}[/tex]
The first equation gives [tex]U=10\sqrt{\frac{G(M+m)}{r_0}}[/tex].
The second equation gives [tex]V_0=9\sqrt{\frac{G(M+m)}{r_0}}[/tex].
One can check easily that the mechanical energy in the second equation is positive:
R.H.S.[tex]=-\frac{GMm}{r_0/10}+\frac{1}{2}\frac{Mm}{M+m}U^2=40\frac{GMm}{r_0}[/tex].
Therefore the trajectory is hyperbola(mechanical energy >0), in the point of view of M .
 
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1. What is included in an Intermediate Mechanics Practice Test?

An Intermediate Mechanics Practice Test usually includes problems related to motion, forces, energy, and momentum. These problems may also involve concepts such as Newton's laws of motion, work and power, and rotational motion.

2. How can I prepare for an Intermediate Mechanics Practice Test?

To prepare for an Intermediate Mechanics Practice Test, it is important to review the relevant concepts and equations covered in class. Practice solving problems similar to those that may appear on the test, and make sure to understand the reasoning behind each step.

3. How is an Intermediate Mechanics Practice Test graded?

An Intermediate Mechanics Practice Test is typically graded based on the number of correct answers. Some tests may also include partial credit for incorrect but logical answers.

4. How much time is typically given for an Intermediate Mechanics Practice Test?

The amount of time given for an Intermediate Mechanics Practice Test may vary, but it is usually around 60-90 minutes. It is important to manage your time wisely and not spend too much time on any one problem.

5. What are some common mistakes to avoid on an Intermediate Mechanics Practice Test?

Some common mistakes to avoid on an Intermediate Mechanics Practice Test include not fully understanding the problem, using the wrong equation or formula, and making careless errors in calculations. It is also important to read the questions carefully and pay attention to units and significant figures.

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