College Physics Questions and Answers

Imagine a space shuttle traveling through space at a constant velocity close to c. As the shuttle passes earth, a previously set-up camera starts broadcasting from earth to the shuttle. Since radio waves travel at the speed of light, the shuttle is receiving a constant transmission feed, assuming the camera is broadcasting 24/7.
Now, from what I have understood of special relativity so far, time will flow slower for the astronaut than for the earthlings. Hence, assuming $v=0.8c$, the astronaut will after 30 years have received a video transmission 50 years long!
Is my reasoning correct, that even though the transmission is live, the astronaut would actually be watching things that happened many years ago, while still receiving the "live" feed, which would be stored/buffered in the shuttles memory, thus making it possible for the astronaut to fast-forward the clip to see what happened more than 30 years after passing the earth?

A local inertial frame of reference rocks. A little later it is said that Lorentz frames rock. Are they identical or is one a special case of the other?

From my quick investigation, the spectrum is based on the Rydberg formula, and with a small change, would lead to
$\frac{1}{{\mathrm{\lambda <!--\; \lambda \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}}=\frac{m_{\mathrm{\mu <!--\; \mu \; -->}}}{m_e}\left(R(\frac{1}{n_1^2}-<!--\; -\; -->\frac{1}{n_2^2})\right)$
where $m_{\mathrm{\mu <!--\; \mu \; -->}}$ is the mass of a muon.
So, taking hydrogen as an example, we would observe similar bands, shifted into the x-ray/gamma range.
Is this correct?

Why 8–15 µm is considered "thermal infrared" if typical room temperature kT is 48 µm?
According to Wikipedia:
"Long-wavelength infrared (8–15 µm, 20–37 THz, 83–155 meV): The "thermal imaging" region, in which sensors can obtain a completely passive image of objects only slightly higher in temperature than room temperature - for example, the human body - based on thermal emissions only and requiring no illumination such as the sun, moon, or infrared illuminator. This region is also called the "thermal infrared"."
However, using $\frac{hc}{\mathrm{\lambda <!--\; \lambda \; -->}}=k_{\mathrm{B}}T$, the temperature range 288–308 K (15–35 °C) is equivalent to 50–46.7 µm, while 8–15 µm is equivalent to 1800–960 K (using the same equation).

This is the common problem of a charged particle moving in a static electric and magnetic field.
Say $\mathbf{\text{E}}=(E_x,0,0)$ and $\mathbf{\text{B}}=(0,0,B_z)$
In the inertial frame of reference, the equation of motion is (1):
$\frac{d\mathbf{\text{v}}}{dt}=-<!--\; -\; -->\frac{q\mathbf{\text{B}}}{m}\times <!--\; \times \; -->\mathbf{\text{v}}+\frac{q}{m}\mathbf{\text{E}}$
We can find equations for $v_x$ an $v_y$ and see that the resulting motion is a circular orbit with a constant drift velocity $v_d=\frac{E_x}{B_z}$.
Surely I should get the same answer if I solve the problem in a rotating frame of reference?
I know that (2):
$\frac{d\mathbf{\text{v}}}{dt}{|}_{Inertial}=\frac{d\mathbf{\text{v}}}{dt}{|}_{Rotational}+\mathit{\omega <!--\; \omega \; -->}\times <!--\; \times \; -->\mathbf{\text{v}};$
If I use Eq. (1) as the LHS of Eq. (2), and choose $\mathit{\omega <!--\; \omega \; -->}=-<!--\; -\; -->\frac{q\mathbf{\text{B}}}{m}$, then I get (3):
$\frac{d\mathbf{\text{v}}}{dt}{|}_{Rotational}=\frac{q}{m}\mathbf{\text{E}};$
How do I obtain a constant drift velocity (as mentioned before) from this? Have I used any formula incorrectly? Does the electric field E=$\mathbf{\text{E}}=(E_x,0,0)$ change form in the rotating frame?

Is there a significant error in using De Broglie's equation for an electron at really high speed?
I was wondering if using the De Broglie equation
$\mathrm{\lambda <!--\; \lambda \; -->}=\frac{h}{p}$
for object traveling at really high speeds would result in a significant error. For example if an object travelled at $0.02c$ would the error be negligible? How can I calculate the uncertainty in the result?

"If the frame of reference is translated or rotated, the vector doesn't change."
Although the length of the vector won't change, the angle that this vector makes with the positive direction of the newly defined x-axis changes, no? Hence, how is it possible to state that the vector doesn't change? Have I misunderstood the basic definition of a vector?

Derivation of Wien's displacement law
I know this might be a silly question, but is it necessary to know Planck's Law in order to show that ${\mathrm{\lambda <!--\; \lambda \; -->}}_{max}\propto <!--\; \propto \; -->\frac{1}{T}$? If you set
$u(\mathrm{\lambda <!--\; \lambda \; -->},T)=\frac{f(\mathrm{\lambda <!--\; \lambda \; -->}T)}{{\mathrm{\lambda <!--\; \lambda \; -->}}^5}$
then
$\frac{\mathrm{\partial <!--\; \partial \; -->}u}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}=\frac{1}{{\mathrm{\lambda <!--\; \lambda \; -->}}^5}\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->\frac{5}{{\mathrm{\lambda <!--\; \lambda \; -->}}^4}f=0$
$\frac{\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}}{f}=5\mathrm{\lambda <!--\; \lambda \; -->}$
But I am stuck here because if I integrate the L.H.S.
$\int <!--\; \int \; -->\frac{\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}}{f}d\mathrm{\lambda <!--\; \lambda \; -->}=\log <!--\; \; -->(f(\mathrm{\lambda <!--\; \lambda \; -->}T))=\frac{5}{2}{\mathrm{\lambda <!--\; \lambda \; -->}}^2$

Direction of momentum given by the de Broglie relation
$p=mv$
where $m$ is the mass of an electron, and $v$ is its velocity. In this case, since $v$ is a vector, it's clear that the momentum will be also a vector.
However if the momentum is a vector quantity (and it is), what is the direction of the electron's momentum given by the de Broglie relation
$$\begin{gathered} p=h/\mathrm{\lambda <!--\; \lambda \; -->} \\ p=\mathrm{\hslash <!--\; \hslash \; -->}k \end{gathered}$$
if the Planck constant $h$ is scalar and the wavelength $\mathrm{\lambda <!--\; \lambda \; -->}$ is also scalar. Similarly the reduced Planck constant $\mathrm{\hslash <!--\; \hslash \; -->}$ is scalar and the wavenumber $k=2\mathrm{\pi <!--\; \pi \; -->}/\mathrm{\lambda <!--\; \lambda \; -->}$ is also scalar.

What is half life decay?

What is nuclear waste half life?

For the hydrogen atom, the Balmer Series is the series of line spectra produced when an election falls from any higher energy level and has n=2 as its final energy state. The longest wavelength is the Balmer Series is 656.7 nm. what was the initial principal energy level (n) associated with this photon

Deriving photon energy equation using $E=m{c}^2$ and de Broglie wavelength
So I was learning de Broglie wavelength today in my physics class, and I started playing around with it. I wondered if it was possible to calculate the energy of a light wave given its wavelength and speed. After rearranging a bit, I plugged it into $E=m{c}^2$, and realized I had found the equation for the energy of a photon that I learned at the beginning of my quantum mechanics unit, $E=hf$
$\mathrm{\lambda <!--\; \lambda \; -->}=\frac{h}{p}$
$p=\frac{h}{\mathrm{\lambda <!--\; \lambda \; -->}}$
$E=m{c}^2$
$E=\frac{p}{c}\cdot <!--\; \cdot \; -->c^2$
$E=\frac{hc}{\mathrm{\lambda <!--\; \lambda \; -->}}=hf$
I have a few questions about this. Firstly, I do not understand how it can make sense to do $\frac{p}{c}$ in this context, because, as I understand it, light has no mass. How can I come to $E=hf$ using the mass of a massless object?
Secondly, as I was doing those steps, I thought I would be calculating the energy of the entire light ray. I now realize I was finding the energy of a single photon. In hindsight, this makes sense, because the energy of the entire light ray must depend on some length value, correct?
This led me to two other questions: do light rays have some finite length? how do I calculate the energy of a light ray, not just a single photon?

Why is a nuclear fusion reactor not currently possible?
Nuclear fission reactors are fairly common, but what technological/physical limitations are currently impeding developing the nuclear fusion reactor? The H-bomb is capable of creating nuclear fusion on earth, so the process of creating nuclear fusion is already understood to some extent, although you are detonating a fission bomb to start the fusion process.
Having a cycle between a fusion and a fission reactor would solve some serious energy problems.

A Newtonian homogeneous density sphere has gravitational binding energy in Joules $U=-<!--\; -\; -->(3/5)(G{M}^2)/r$, G=Newton's constant, M=gravitational mass, r=radius, mks. The fraction of binding energy to gravitational mass equivalent, $U/M{c}^2$, is then (-885.975 meters)(Ms/r), Ms = solar masses of body, c=lightspeed.
This gives ratios that are less than half that quoted for pulsars (neutron stars), presumably for density gradient surface to core and General Relativity effects (e.g., billion surface gees). Please post a more accurate formula acounting for the real world effects.
Examples: 1.74 solar-mass 465.1 Hz pulsar PSR J1903+0327, nominal radius 11,340 meters (AP4 model), calculates as 13.6% and is reported as 27%. A 2 sol neutron star calculates as 16.1% and is reported as 50%. There is an obvious nonlinearity.

What is smallest wavelength line in the Balmer series:
A) 364.5 nm
B) 402.5 nm
C) 388 nm
D) 656 nm

Calculate the wave number for the second line of the Palmer series

The SNF accounts for 99.9% of the mass of an atom.
So if 4 hydrogen atoms become 1 helium atom during nuclear fusion does that change the proportion of SNF as we have 4 SNFs becoming 1 SNF and 12 particles becoming 6 particles. Is it now 99.8% or something?

I'm trying to write a paper arguing for why we should spend more money on nuclear fusion research. I keep reading that "nuclear fusion produces no long lived radioactive waste", but when I looked up why this is the only good source I could find was Quora, clearly not something I can really cite. So, I was hoping someone here could explain it to me, and hopefully also leave their sources so I could use them as well.