Differential Equation involving mechanics

• May 12th 2010, 10:10 AM
cooltowns
Differential Equation involving mechanics
Hi

I need help with the following question. I have attached my question. I need help with part b)

this is what i have managed to do so far, but i don't think i am doing it correctly.

$m\ddot{r}=\frac{mMG}{r^{2}}$

dividing both sides my $m$

$\ddot{r}=\frac{MG}{r^{2}}$

$\int\ddot{r}dt=\int\frac{MG}{r^{2}}dt$

$\dot{r}=\frac{MG}{r^{2}}t+C
$

any help is appreichated

thanks

(Happy)
• May 12th 2010, 12:04 PM
chisigma
This problem can be solved as in...

http://www.mathhelpforum.com/math-he...tml#post510114

Setting for semplicity sake $M G = \mu$ the DE becomes...

$r^{''} = -\frac{\mu}{r^{2}}$ (1)

Now because is...

$r^{''} = \frac{d r^{'}}{dt} = \frac{d r^{'}}{dr} \frac{d r}{dt} = r^{'} \frac{d r^{'}}{dr}$ (2)

... we obtain from (1)...

$r^{'} dr^{'} = -\frac{\mu}{r^{2}} dr$ (3)

... i.e. a separable variables DE the solution of which is...

$r^{' 2} = \frac{2 \mu}{r} + c_{1}$ (4)

... just like in Your textbook. The further step could be finding $r(t)$... a little less confortable step (Thinking) ...

Kind regards

$\chi$ $\sigma$