- #1
Ackbach
Gold Member
MHB
- 4,155
- 92
Here is this week's POTW:
-----
In freshman physics, or high school physics, you analyzed free-fall motion by assuming that the gravitational force was a constant, $mg$, from which you can use the constant-acceleration kinematic equations. Let's improve on this approximation one notch by introducing a more accurate gravitational force:
$$F=\frac{GM_{e}m}{r^2},$$
Newton's Universal Law of Gravitation. Here $G$ is the gravitational constant, $M_e$ is the mass of the earth, $m$ is the mass of the object in the next sentence, and $r$ is the distance from the center-of-mass of the Earth to the center-of-mass of the object. Assuming that the object starts from rest at some initial distance $r_0$ from the center of the earth, find its position as a function of time. You may neglect air resistance, the dependence of pressure on height, the Coriolis effect, etc. That is, for this model, the only force present is gravity.
[EDIT] You may obtain an implicit result.
-----
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
-----
In freshman physics, or high school physics, you analyzed free-fall motion by assuming that the gravitational force was a constant, $mg$, from which you can use the constant-acceleration kinematic equations. Let's improve on this approximation one notch by introducing a more accurate gravitational force:
$$F=\frac{GM_{e}m}{r^2},$$
Newton's Universal Law of Gravitation. Here $G$ is the gravitational constant, $M_e$ is the mass of the earth, $m$ is the mass of the object in the next sentence, and $r$ is the distance from the center-of-mass of the Earth to the center-of-mass of the object. Assuming that the object starts from rest at some initial distance $r_0$ from the center of the earth, find its position as a function of time. You may neglect air resistance, the dependence of pressure on height, the Coriolis effect, etc. That is, for this model, the only force present is gravity.
[EDIT] You may obtain an implicit result.
-----
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!