## anonymous one year ago how to solve this ODE: I will post it

1. anonymous

$\frac{ d ^{2} x }{ dt } = 32 - \frac{ v^2 }{ 8 }$

2. anonymous

I tried to solve it for v then for x, but it was very messy.

3. anonymous

@e.mccormick

4. anonymous

@phi

5. anonymous

$\frac{ d }{ dt }\left( \frac{ dx }{ dt } \right)=32-\frac{ v^2 }{ 8 }$ $\frac{ dv }{ dt }=\frac{ 256-v^2 }{ 8 }$ $\frac{ dv }{ 256-v^2 }=8 dt$ $\frac{ dv }{ \left( 16-v \right)\left( 16+v \right) }=8 dt$ $\left\{ \frac{ 1 }{\left( 16-v \right)\left( 16+16 \right) } +\frac{ 1 }{ \left( 16+16 \right)\left( v+16 \right) }\right\} dv=8dt$ can you solve further?

6. anonymous

of course you don't need to make partial fraction as you can sub.with sin theta and you will get ln(sec+tan) +k and then integrate which that!! my text didn't discuss DE at all just basic integral, but he gave that equation and said after 1 minute it will hit the ground, so find height??(ans 555) I am puzzled about what is his simple way to deal with that?

7. anonymous

when t=1 sec. when it touches the ground v=0 find the value of constant then again integrate to find x when t=1 x (height)=0 find value of constant.

8. anonymous

sorry 1 min. just take t=1

9. IrishBoy123

@Catch.me why don't you just scan or link the actual question? the DE is not the problem, it is quite straightforward. sight of the question and the IV's would be interesting. @phi

10. IrishBoy123

and, as a complement to @surjithayer 's excellent input: $\ddot x = \frac{dv}{dt} = \ v \frac{dv}{dx}$ :p

11. anonymous

V final can't be zero as it is a different scenario. I solved it, but 60 in expm. worries me. any way thanks guys