Homework Statement
Show that I = log(u)-u+2log(v)-v is an invariant of the following system
\dot{u}=u(v-2)
\dot{v}=v(1-u)
Homework Equations
The Attempt at a Solution
The question was given on a homework assignment, but I have very little idea what it is asking for and even...
So then would the derivative be
\frac{d^2x}{dt^2t}=\frac{d}{dt}\frac{dx}{dt}=\frac{d}{dt}(v_0 \frac{dX}{dT})=\frac{d}{dT}(\frac{g}{v_0})(v_0 \frac{dX}{dT})
\frac{d^2x}{dt^2}=g\frac{d^2X}{dT^2}
similarly
\frac{d^2y}{dt^2}=g\frac{d^2Y}{dT^2}
Thanks for your help
Homework Statement
Consider a projectile subject to air resistance. The drag force is F=-\hat{v} C v^{a} where C and a are constants and v=\dot{r}. Restrict the problem to 2-D and take y in the vertical direction. Take the magnitude of initial velocity to be v_{0} and the initial position to...
Homework Statement
Find the first 3 (non-zero) terms of the asymptotic series as x→∞ for ln(ex+1)
Homework Equations
ln(1+ε) ~ ε-ε2/2+ε3/3-O(ε4) (x→0)
The Attempt at a Solution
I found
x+ln(1+e-x) (x→∞) = ln(ex+1) (x→∞)
ε=e-x
which allows me to use the Maclaurin series...
I'm sorry, I don't know heat balance equations. I've looked around, but I'm not sure which source to go by. Also I wouldn't know what to put for a change in temperature. Additionally, in equilibrium, I calculated that the pressure for this system is 649.3 atmospheres, and I don't know if any...
Homework Statement
Supercooled water at minus 5C in a thermally insulated container suddenly transforms into an equilibrium ice-water mixture.
a. What fraction of water has frozen?
b. How much was the entropy change per gram?
Homework Equations
Clapeyron Equation...
perhaps I am supposed to assume that the gas is monoatomic? My textbook is vague and its not clear to me whether all ideal gasses are considered monoatomic.
this is what i found:
\gamma= 5/3 (ideal monatomic gas).
Homework Statement
Suppose that one mole of an ideal gas expands in a quasi-static adiabatic process from P1 = 1 Pa and V1 = 1m3 to V2 = 8m3. What is the change in the pressure and the entropy of the gas?
Homework Equations
PV\gamma=constant
The Attempt at a Solution
I can't come...
I'd just like to check my answer, it seems too big
Homework Statement
Air initially at 20◦C is compressed by a factor of 15.
What is the final temperature assuming that the compression is adiabatic and \gamma ≈ 1.4 the value of \gamma for air in the relevant range of temperatures? By what...