Homework Statement
If a and b are both quadratic residues/nonresidues mod p & q where p and q are distinct odd primes and a and b are not divisible by p or q, Then x2 = ab (mod pq)
Homework Equations
Legendre symbols: (a/p) = (b/p) and (a/q) = (b/q)
quadratic residue means x2 = a (mod p)
The...
Thanks. That is a lot easier to integrate. After integrating, I get
v2=\frac{mg}{k}-ce^{\frac{2kx}{m}}
with initial value V(0)=0, I get c=\frac{mg}{k}
so v=\sqrt{\frac{mg}{k}\left(1-e^{\frac{2kx}{m}}\right)}
x would have to represent the height fallen, and it will have to be...
With the initial condition: v(0)=0, c=1 and
e^{ln\sqrt{1+\sqrt{\frac{k}{mg}v}}}e^{-ln\sqrt{1+\sqrt{\frac{k}{mg}v}}}=e^{\frac{t\sqrt{kmg}}{m}}
This gives me velocity if I know time, but I know height instead. How do I figure that into my problem?
Ok. I see why. I also now see why I should have 1/\sqrt{kmg} in front instead of what I had.
So I now have \frac{1}{\sqrt{kmg}}\left(ln\sqrt{1+u}-ln\sqrt{1-u}\right)=\frac{t}{m}+c
\left(ln\sqrt{1+u}-ln\sqrt{1-u}\right)=\frac{1}{\sqrt{kmg}}\left(\frac{t}{m}+c\right)...
So this gives me
\int\frac{dv}{mg-kv^{2}} = \int\frac{dt}{m}
with u substitution:
\sqrt{\frac{k}{mg}v}(\frac{1}{2}\int\frac{du}{1+u} + \frac{1}{2}\int\frac{du}{1-u}) = \int\frac{dt}{m}
becomes
\sqrt{\frac{k}{mg}v}(\frac{1}{2}ln\left|1+u\right| + \frac{1}{2}ln\left|1-u\right|) =...
After thinking more, differentiating with respect to height would not change my equation (except from t to h), but my initial value would be different for each height. So I would have a different specific solution for each height.
I'm still curious as to whether I solved the equation...
If mg, and k =1, then I know I would get arctanh v. It's been a while since calc, so I'm not sure what to do with the constants.
My other thought is to go the natural log route:
ln(mg-kv2)=t/m+c
then raising both sides to e:
mg-kv2=cet/m
using v(0)=0 as my initial condition, c=180...
Homework Statement
Find the velocity of a 180 lb object falling from a state of rest from 5 ft, 15 ft, 25 ft, and 45 ft
Homework Equations
using the formula f=mg-Kv2, where g is the acceleration due to gravity, m the mass of the body, v its velocity. The terminal velocity is 120mph...
Thanks! I got y=x^2/(c-x)
Every time I use the x2 button at the top I get SUP/SUP
I thought I also had to plug it back into the original to verify, but I found out solving for y is good! :)
Homework Statement
Consider the equation (y^2+2xy)dx-x^2dy=0 (a) Show that this equation is not exact. b) Show that multiplying both sides of the equation by y^-2 yields a new equation that is exact. C) use the solution of the resulting exact equation to solve the origional equation. d)...