- #1
Potatochip911
- 318
- 3
Homework Statement
The free fall acceleration of a mass ##m## above the Earth's surface in one dimension can be represented by ##m\ddot{y}=-\frac{mMG}{y^2}## where ##M## is the mass of the Earth and ##G## is the gravitational constant. With ##\dot{y}(t=0)=0## and ##y(t=0)=y_0##.. (1) Find an equation for ##y(t)##, then ##y(t)## can be represented by the equation ##y(t)=R+\alpha(t)## where ##R## is the radius of the earth. (2) Solve the differential equation in (1) using this, that ##\alpha_0## and ##\alpha(t)## are ##<<## than ##R##, ##y(t=0)=R+\alpha_0## and the taylor expansion ##f(t)=f_0+tf'(t)##
Homework Equations
3. The Attempt at a Solution [/B]
The first part of this question is quite trivial, multiplying the first equation by ##\dot{y}## and cancelling common factors then integrating we obtain $$\int_{0}^{\dot{y}}\dot{y}dy=-MG\int_{y_0}^{y}\frac{1}{y^2}$$
which after solving for ##y(t)## results in $$y(t)=\frac{2MGy_0}{\dot{y}^2y_0+2MG}$$ Continuing on to the second part of the question now we have ##R+\alpha(t)=\frac{2MGy_0}{\dot{y}^2y_0+2MG}##, note that since ##y(t)=R+\alpha(t)## applying the initial conditions we obtain ##\alpha(t=0)=\alpha_0## and taking the derivative we find that ##\dot{y}=\dot{\alpha}(t)\Rightarrow \dot{y}^2=\dot{\alpha}(t)^2## which gives us and then using the relation ##y_0=R+\alpha_0## $$R+\alpha(t)=\frac{2MGy_0}{\dot{\alpha}^2y_0+2MG}=\frac{2MG(R+\alpha_0)}{\dot{\alpha}^2(R+\alpha_0)+2MG}$$
Now I'm pretty sure I can get rid of the ##\dot{\alpha}^2\alpha_0## term on the denominator since it should be negligible compared to the other terms. I'm quite confused as to what to do with the taylor expansion though since I don't understand why we are using the Maclaurin series expansion (my professor told us to use this) when we haven't defined that ##t\approx 0##, regardless, if I plug in the expansion and remove the ##\dot{\alpha}^2\alpha_0## term I obtain $$R+\alpha_0+t\dot{\alpha}=\frac{2MG(R+\alpha_0)}{\dot{\alpha}^2R+2MG}$$
After some tedious simplification I arrived at $$\dot{\alpha}^2R^2t+\dot{\alpha}(R^2+\alpha_0R^2+2MGt)=2M\alpha_0(1-G)$$
Which just seems way too complicated (I have no clue how I would go about solving this) so I'm assuming I have made a mistake somewhere, if I had to guess I would say it has to do with the taylor expansion.