dx/dt =1, x(0,s)=0, dy/dt=x, y(0,s) = s, du/dt=(y-1/2x^2)^2, u(0,s)=e^s
I did well at the beginning to get x(t,s) =t and y(t,s)=1/2t^2 + s, but got stuck with the du/dt part.
You can sub in x=t and y=1/2t^2 +s for x and y to get du/dt = s^2. But that's still three variables, and I can't see...
I was thinking about ballistic pendulums and the symmetry they exhibit. In the simplest case, you have one ball that begins at a certain height and collides with another ball at rest. You can calculate via conservation of momentum and energy the new velocities and max vertical displacements...
While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...
I seem to be getting an unsolvable integral here (integral calculator says it's an Ei function, which I've never seen). My thought was to use Bernoulli to make it linear and then integrating factors. Is that wrong? The basic idea is below:
P(x) 1, Q(x) = 1/2(1-1/x), n=-1, so use v=y^1-...
It makes sense that a=2 would cause problems because then we wouldn't have a matrix of full rank and we'd be unable to determine a value for w. But the key also says that when b+4a^2-4a-7≠0. Why is that an issue? For example, if a=1, that just says implies that w=0. Through back-subsitution...
I'm confident in calculating the beginning values. Only question I had was whether in this graph y2 is assumed positive or negative. Obviously f(pi)= negative value, but is y2 negative or is it = |f(pi)|. Assuming the latter, it all makes sense and when I sum the two Simpson's Rule estimates...
I'm pretty confused here because after getting stuck on this problem, I tossed it into an integral calculator and it said the answer was 2 Si(x) + cos(x)/x + C. In intro calc we definitely haven't learned the Si(x) function or even gotten to any of the Taylor polynomial stuff yet.
I tried IBP...
I was told this problem could simply be solved with calc-1 techniques, so I'm tempted to say we could do d/dx(x∫(limits: 0,x) sin(t) dt. Then it's a simple product rule: d/dx (x) * ∫(0,x) sint dt + x * d/dx(∫ (0,x) sin (t) dt) = 1 - cos(x) + x*sin(x). However, I wonder if we have to allow that...
I found that f= x -2yz. To maximize f, I can first inspect the solutions to grad(F)=0. z=y=0 pops out, but I'm not sure what to do with the x-component equaling 1. Do we just include (x,0,0) as a solution? I think the problem wants specifics though, based on what I've seen previously from...
The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative.
I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to...
Problem: See Attachment. Parts (a) & (b) are clear, but my confusion arises in (c)-- I feel like there is a much simpler form. While technically my answer is correct, there must be something I'm missing.
I parameterized the curve C=(t, e^2t, e^2t) and got c'(t)=(1,2e^2t,2e^2t), which should...
Real World Application here. I'm creating a virtual meeting involving six people and will have the first round include two groups of three. Then we'll switch a few times. I tried hopelessly to do it such that after three rounds, everyone would see everyone else at least once. Was so close...
I'm all messed up on this problem. I see you can get the solution (74cm, as listed in the back of the book) by simply setting mgh=1/2kx^2, saying that h=x, and then adding 15 cm to that since that's the original length of the spring. This is the solved solution I was given. But now I think...
I'm confused about what happens during exercise or after sympathetic nervous system stimulation. I've read that the blood vessels going to the skeletal muscles constrict to increase BP and get blood moving faster, but also that the blood vessels going to the GI constrict to shunt blood flow...
I’m kind of stuck on this one. I don’t see how we can use the acceleration of an elevator to find anything about the value of g. We can of course get the acceleration of the elevator, but it feels like that would be independent of gravitational strength. I feel like we need the mass or the...
OK, so I'm trying to work out a few ideas regarding locality. I've studied at the undergrad level in the past (including quantum), but with professors that slaved away at proving math constructs and never bothered to indulge in clarifying the context of any concepts, so I'm pretty weak here...
Homework Statement
One of the reasons a single photon could not produce an odd number of electrons and positrons is
a) energy would not be conserved
b) unattainable photon energies would be needed
c) matter would be created
d) charge would not be conserved
Homework Equations
The Attempt at...
Homework Statement
A long board is free to slide on a sheet of frictionless ice. A skater skates to the board (laid horizontally relative to the skater's motion) and hops onto one end, causing the board to slide an rotate. In this situation, are angular and linear momentum conserved...
The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length.
I began...
I'm a little confused by what exactly determines if a reaction is reversible or irreversible. I thought I have heard that if the delta G of a reaction is negative (which means it's spontaneous), that it's then irreversible, but that doesn't make a whole lot of sense when I think about it...
Homework Statement
A car with a mass of 1000 kg crashes into a rubber wall with an initial velocity of 30 m/s and recoils at 15 m/s. The change in momentum occurs over 1 meter. Find the force that the car experiences and time for which this force acts.
Homework Equations
v22 - v12=2ad
F=ma...
I have been working with equations throughout the semester and using them to find cutoff frequencies and such, but when you say something like TE10, TE20, TE21, etc., I am realizing that I am having trouble getting a physical idea of what exactly these things are. Of course if you transmit at a...
Now w/ Attachment: Telephone Wire w/ Load and Characteristic Impedance
Homework Statement
Please see attachment.
Homework Equations
I(z)=I+(exp(-γz)) + I-(exp(γz))
V(z)=V+(exp(-γz)) + V-(exp(γz))
The Attempt at a Solution
I solved easily for the characteristic impedance and gamma...
I was wondering, is there a formula that converts a given z-score to its respective percentile rank? I know I can look up the conversion in a table, but I have a lot of data, and would rather just program a formula into Excel. Obviously, there is some sort of way that whoever created the table...
I know computer programming skills are important in applied physics/engineering jobs and am looking for some advice as to how and what I should do to teach myself these skills. I am in a Photonics Master's degree program, which gears students toward acquiring jobs in industry (semiconductors...
Homework Statement
Evaluate the matrix element <U210|z|U100> where by |Unlm> we mean the hydrogen atom orbital with it's quantum numbers.
Homework Equations
The Attempt at a Solution
So where I'm getting stuck is on the integral, because the "U" portion of the wave function is...
In my lecture notes, there is a slide about an external field interacting with a system of electron spins. In this situation can we assume that B=B_0 ( or in other words that μ_0*M =0)? I would think so, but does a system of electron spins imply that these electrons belong to some matter...
Homework Statement
Consider the energy surface
E(k)=h2((kx2 +ky2 )/ml+kz2/mt
where m_l is the transverse mass parameter and m_l is the longitudinal mass parameter. Use the equation of motion:
h(dk/dt)= -e(vXB) with v=∇k(E)/h to show that ωc=eB/(ml*mt)1/2 when the static magnetic field B...
After looking at Umklapp scattering, I believe I have finally gotten most of it down, but a few things are still not clear.
1) Momentum is not conserved for certain phonon collisions, and a certain number of reciprocal lattice vectors are transferred to the crystal lattice...
Hello, I am having quite a bit of trouble really grasping Brillouin Zones and their relation to phonons, energy propagation, etc. I've got a few questions, and there will probably be a number of misconceptions, but I figure they'll clarify what I exactly don't understand. I think a lot of the...