Friday, September 23, 2022

Stellar Luminosity Problem Solutions

 (1) The apparent V-band (filter) magnitude of a star is 8.72, and it requires a bolometric correction of -0.48. Find the apparent bolometric magnitude of the star. (Hint: Apparent bolometric magnitudes are obtained in an analogous way to the absolute forms)

Solution:

1) mV= 8.72

m bol * = mV + B.C. = 8.72 + (-0.48) = 8.24

Hence, m bol *  = +8.24

(2) A star has a color index (B - V) of +1.0 and its apparent B-band magnitude is 6.4. The corresponding bolometric correction is - 0.5. Find the apparent bolometric magnitude of the star.

Solution:

mB = 6.4 and (B - V) = (mB - mV) = 1.0

Then: mV = mB - 1.0 = 6.4 - 1.0 = 5.4

m bol *  = mV + B.C.

m bol * = 5.4 + (-0.5) = +4.9

(3) The star Alhena in the constellation Gemini is at a distance of 30 pc. If it has (B - V) = 0.00, and mV = +1.93, find the apparent B-band magnitude, mB.

Also find the absolute visual magnitude and the absolute bolometric magnitude of the star.

Find the luminosity of Alhena in terms of the solar value.

Solution:

(B - V) = mB - mV = 0.00

Then if mV = +1.93, then mB = +1.93.

The absolute visual magnitude MV = mV - 5 log D + 5

D = 30 pc, so:

MV = 1.93 + 5 log (30) + 5 = 1.93 - 5(1.477) + 5

MV = 1.93 - 7.39 + 5 = -0.46

The absolute bolometric magnitude is:

Mbol* = MV + B.C.

where B.C. = 0.72 (from the table) so:

Mbol* = -0.46 + (-0.72) = -1.18

The Luminosity L' is obtained using:

Log (L'/L) = 0.4 (Mbol - Mbol*)

where Mbol (Sun's) = 4.63

Log (L'/L) = 0.4 (4.63 - (-1.18)) = 0.4 (5.81) = 2.32

Antilog (2.32) = 209 approx.


L'/L = 209 and L' = 209 L   or  209 times the solar luminosity.


Is U.S. Really Suffering From A Mental Health Crisis - Or A Political Crisis AFFECTING Mental Health?

 





For the first time ever-  the U.S. Preventive Services Task Force has recommended anxiety disorder screening for U.S. adults in the primary care setting, and even with no symptoms. Anxiety disorders are often unrecognized and under-detected in primary care: One study cited by the task force found the median time for initiating treatment for anxiety is a staggering 23 years.   (The task force said there wasn’t enough evidence to determine the accuracy of screening tools in older adults, which may not be sensitive enough to distinguish between anxiety symptoms and conditions of aging.)

Global prevalence of anxiety and depression increased by 25 percent during the first year of the pandemic, the World Health Organization reported earlier this year. By the end of 2021, the WHO said, “the situation had somewhat improved but today too many people remain unable to get the care and support”. 

Meanwhile,  a study published in 2021 in The Lancet estimated that the pandemic had caused an additional 53.2 million cases of major depressive disorder and 76.2 million cases of anxiety disorder globally. Less well known is that referrals for psychosis have increased as much as 29 percent and shows no sign of stopping as political division,  animus - driven by lies, QAnon and Trumpism- continue to ramp up.   (Psychosis involves seeing or hearing things that other people don’t (hallucinations) and also developing beliefs that aren’t based on reality, such as Democrats being a ring of pedophiles who sacrifice children to Satan).

But the Covid -19 pandemic, has also been fingered with one health official saying outright: 

"Covid has taken a tremendous toll on the mental health of Americans,

However, the overuse of social media must also be reckoned as a factor, especially as lockdowns proceeded during the first year of the pandemic.  People then depended excessively on their devices for support and social interaction with negative results on many mental health fronts.  

We know, based on earlier research, that brains take a mega hit when locked into cell phones, their assorted apps, and other devices.  Look no further than the sort of idiocy recently portrayed on Tik-Tok, cooking chicken in Nyquil.  What sounds 'cool' is actually a deadly form of social stupidity given the amount of acetaminophen in the latter is enough to permanently damage the liver.  This, of course, is as lamebrained as the foolishness of  teens challenging peers to eat TIDE laundry pods some five years ago, e.g.

Tide pod challenge: What is it, and why is the viral video dare dangerous? - The Washington Post

These may seem like frivolous examples to some, but the exposure to such videos creates impacts on the developing brain.  We already know, based on research conducted at the Univ. of California - San Francisco  (UCSF) some years ago - that even a transitory anomalous stimulation can extensively alter neural circuits.  A kid who partakes in a stupid social media dare or stunt - say on TikTok- can then have his sensory cortex sufficiently reorganized as to create a new mental map.  A kid who merely views such a video can suffer the same fate. In the words of one UCSF researcher: 

"The long term effect  on the quality of intellectual life can be deadly.'  

The effect on the intellectual apparatus of adults is no less severe, especially in terms of attachment to certain politically oriented social media sites.  Sites like 4chan, 8chan, 'The Donald' and Trump's Twitter knockoff are all prone to spread QAnon lies, psychotic conspiracy ideations, and Trumpian aspirations to political violence.  Again, the very sight of others at rallies chanting allegiance or giving hand salutes can re-order the brains of vulnerable viewers already open to mischief, lies and disinformation.

Depending on the mainstream media to filter facts from outpourings is no use, since as WaPo columnist Jennifer Rubin has written,  the media can't see beyond its false equivalence trope.  As she observed: 

The cynical GOP leaders who know that Trump is unfit for office and that many of his cult followers have become violent should not be treated as ordinary party hacks. They are enablers of a dangerous movement. Yet they continually evade persistent, aggressive questioning.

 Compare this with the mainstream media’s response to Biden’s recent speech condemning the MAGA movement. Yet many in the mainstream media turned up their noses. “Biden should have been more welcoming,” they said. “He’s too divisive!

It seems to be very deep in the mainstream press’s DNA to strain for equality when none exists,” said Margaret Sullivan, media critic and author of the upcoming memoir, “Newsroom Confidential: Lessons (and Worries) From an Ink-Stained Life.” She adds, “Maybe journalists just don’t have the language to truly get across how disturbing and abnormal some of this stuff is. If so, it’s high time to grapple with that.”

In light of the mainstream media's total incompetence the combination of social media exposure and what is called reification, cannot be discounted.  (Reification is the process by which the effects of a political arrangement of power and resources start to seem like objective, inevitable facts about the world.)  QAnon  psychotic drivel reinforced by Trump at his rallies- like the one in Ohio, e.g. 

by David Badash | September 20, 2022 - 7:16am | permalink

 Then has the effect of reification to make the adherents and participants actually believe what he says has objective reality. This kind of dangerous twaddle is then reinforced on Trumpian -friendly social media with the result that adherents' mental maps are re-arranged as well as their cognitive sensory apparatus overall.  As our German friend Marlies observed, "This is not too different from Hitler's rallies in Nuremberg."  (Marlies was alive at the time of Hitler's reign, and saw her nation descend into madness in the 1930s.)   

The real danger is via repetition of the lies and nonsense. not only on degenerate sites like 8chan and 'The Donald" but by the likes of Trump and his GOP knob polishers (e.g. Lindsey Graham) as well. Like when Trump asserted night before last on Hannity that he had the power to declassify documents merely from thought. Repeated often enough tens of millions of his QAnon and other zombies will believe it owing to a phenomenon called the 'Illusory truth effect' whereby misinformation repeated often enough becomes the truth.  Of course, Nazi Propaganda minister Joseph Goebbels knew this as early as 1931 - see his famous quote about lies at the top.

The near total failure of our mainstream media to make things clear, i.e. call out Trump and GOP lies for what they are. means millions are in danger of being hostage to reification.  This presents a political source for mental anxiety and disorder in an already hyper-polarized nation, as opposed to evidence for a specific mental illness.  In this case one must be wary of the tendency toward Medicalization. 

Medicalization shifts and distorts the language by which we try to figure out what caused a problem, and what can be done to fix it. It generally puts the focus on the individual as a biological body, at the expense of factoring in systemic and structural conditions.  In a way it is similar to the fundamental attribution error from psychology. defined as:

The tendency to blame individuals for their level of failure or success without considering the aspects of the social structure that impel or impede their progress. 

Examples of psychological "medicalization" were abundantly obvious in the Diagnostic and Statistic Manual (DSM) - V, where one beheld such chestnuts as:

  Intermittent Explosive Disorder (or IED) - "recognized by bouts of explosive anger out of all proportion to the cause."


- Temper Dysregulation Disorder with Dysphoria - a new definition "for all children with outbursts of temper", i.e. any brat or rug rat pitching a temper tantrum in a store because mommy won't give him a Snickers bar

The point is that the process of medicalization alters the basis of a real problem from a politically systemic to a medical one by the use of clever language manipulation. Treating the root cause of the “epidemic” of diabetes effectively, for example, would  require serious changes to the available diet and activity levels of a population, not by slinging medications (like Metformin) at it.

Treating the current "mental health" crisis requires an end to the political violence, propaganda, and discrimination that amplifies the probability that an individual will develop a given psychological diagnosis, from anxiety disorder to depression to schizophrenia.

In the case of the current U.S. anxiety and depression epidemic we are faced not with a serious medical issue or problem, but rather a crisis in our politics (dysfunctional political system) which has adversely affected the mental health of millions.  That is why people are lined up to get appointments at therapists' offices, even via tele-meets.

The failure of the mainstream media to be honest and forthright about our deviant political system means a prop for stability, recognition and self-reference is absent.  This has  allowed too many to be gaslighted into questioning their own mental balance. The more vulnerable then risk becoming FOX followers or QAnon adherents for the sake of preserving personal identity.   Thus we have really a crisis affecting mental health as opposed to a mental health crisis. The unstable, hostile and divisive political environment - especially now with 40,000 poll watchers being recruited by the GOP to scan voters at the polls -  invites incalculable stress which triggers diverse psychopathologies.

This crisis is compounded via  an added brain dysfunction - traced to over exposure to media which endlessly repeat the lies.  The cumulative effects of the disinformation barrage - often daily- then dramatically change mental maps and neural connections in adverse ways.

Among the foremost real ill effects is that personal attention spans are devastated.   A Neilson research finding some years ago was that the average American spent 56 seconds glancing at a web page.  Deeper reading for understanding, say from appropriate books and news sources, is almost unheard of.   Instead of neural links being forged and strengthened they've become frayed and fragile.  As my now tenure track Psychology niece Shayl puts it: 

"I can confidently say that up to 90 percent of our mental health issues are directly traced to time online and the inability to separate fact from political fictionThank God for websites like your blog!"  

But my blog  (or similar ones) should not be the only oases in a fact-challenged news media desert landscape. And, truth be told, if more of the mainstream media did its job my blog would be redundant. We'd already have sources that 'tells it like it is' as opposed to saying what people want to hear.  Shayl, not surprisingly, checks  my blog content each day to see if there's something she can use and writes back "in almost every case there is, even in the stuff to do with economics or Covid."  Well, I appreciate her vote of confidence - even if I see attention somewhat slipping in reads of political posts - as we approach the midterm election.

But I trace that to the difficulty of sustaining the mental discipline to read difficult (and even politically challenging) posts in an environment replete with political division, toxicity and yes- reification.  After all, to quote one Cornell psychologist: "The Web never encourages us to slow down.  It keeps us in a state of perpetual mental locomotion."

See Also:

by Thom Hartmann | September 22, 2022 - 7:56am | permalink

— from The Hartmann Report

Excerpt:

Donald Trump has built a cult around himself. This is dangerous to America and dangerous to democracy.

Cults of personality in governance are broadly incompatible with democracy. They usually erupt in dictatorships where the Great Leader’s face and sayings are splashed all over public places. Think Mao’s China, Stalin’s USSR, Hitler’s Germany, Kim’s North Korea.

On a smaller scale and in a different context, we see how destructive such personality cults can be with the deaths around Jim Jones’ Jonestown, David Koresh’s Branch Davidians, and Charles Manson’s Family.

And:

by Will Bunch | September 22, 2022 - 7:38am | permalink

— from the Philadelphia Inquirer

Excerpt:

If you are a political fanatic, you've surely heard the old saying that when fascism comes to America, it will come wrapped in the flag and carrying a cross. That's been proven true in this fraught year of 2022 as Christian nationalism rises to our extreme right, but no one predicted this:

That U.S. authoritarianism would also come with a bat-guano crazy musical soundtrack—music that sounds like a Bible Belt altar riff but is actually tied to the weirdly popular QAnon conspiracy theory whose legion of followers believe there's an elite global cabal of child-trafficking, baby-blood-drinking liberal politicians and movie stars.

This terrifying crossing of some kind of autocratic Rubicon happened—where else?—at a Donald Trump rally in Youngstown, Ohio, on Saturday night.


And:

by Amanda Marcotte | September 21, 2022 - 7:32am | permalink

— from Salon

Excerpt:

It's telling how normalized events like this have become; they barely make ripples now in the national news. On Monday, Tucker Carlson of Fox News used his massive cable news audience — an average of over 3 million viewers a night — to spread a QAnon-style conspiracy theory and encourage people to commit violence against teachers and health care providers. Media Matters collected the clip. It's hard to follow for those who aren't already well-versed in the right's mythology accusing LGBTQ people and their allies of "grooming" children to be victims of sexual abuse. But what is undeniable is that Carlson is calling for violent vigilantism in response to his false accusations against doctors and teachers.

And:

by Thom Hartmann | September 21, 2022 - 7:01am | permalink

— from The Hartmann Report


Excerpt:

Last week a very reasonable sounding fellow called into my show and said, essentially, “I voted twice for Trump and would again. Why do you think I’m a threat to America?”

We ended up debating tax policy and I never did fully answer his question (I’m a sucker for “reasonable”), so here’s my shot at it.

*Trump and the GOP are endorsing Tucker Carlson’s “Great Replacement Theory” that argues, in most of its various iterations, that wealthy international Jews (“George Soros”) are collaborating with Black and Hispanic people to replace white people at all levels of society.

This has led to an explosion of antisemitic, anti-Black, and anti-Hispanic violence and the murder of numerous people, both Black, Hispanic, Asian, and Jewish, in venues from synagogues to supermarkets to churches.

Wednesday, September 21, 2022

The Federal Reserve Governors Need To Hike The Short-Term Fed Rate To At Least ONE Percent Today

 

               It's now on Jay Powell to be the adult in the room at Fed Meeting.


"That we Americans allot the richest rewards of our economy to speculators, is a question of mores; that we allow one-sixth of our fellow citizens to be ill-housed, ill-clad, and ill-nourished is a question of morals. That we shrink from our problems instead of attacking them with eagerness, generosity and hope is a question of morale. The questions are obviously interrelated." - George P. Brockway, in 'The End of Economic Man - Principles of Any Future Economics', p. 86.


Megan Naquin, quoted in a WSJ piece yesterday (Food Prices Are Getting To People ,  p. A11) said she now spends more on groceries for her family of eight than on her $1,500 monthly mortgage.  In her own words:


Stuff hasn’t just gone up a little bit, it’s gone up tremendously!"


Meanwhile, we learn that when asked about her recent grocery trip,  single mom Jamaica Bonvillian teared up and blurted out: "I just get to the point where I break down and I’m like, ‘OK, how do I do it?"


This is the case now given:


"Grocery prices were 13.5% higher in August compared with a year earlier, according to the Bureau of Labor Statistics. That is the biggest 12-month jump since March 1979. With no relief in sight, many consumers say they are struggling to keep up."  


This inflation 'cliff' - with millions at their wits' end trying to figure out how to cope - is why Federal Reserve Chairman Jay Powell needs to stand firm today and deliver a 1 percent interest rate hike.  


The markets, we are informed “don’t like it”.  Why not? Hell’s bells, the Fed has been feeding Wall Street cheap money for over ten years now. Thus, Wall Street's speculators ('margin' aficionados)  have been slow to come around to the Fed’s new posture, because it is at odds with how the Fed has acted for years.  Why would these cheap money crack heads expect anything different?


Recall up to the late fall of 2019 the Fed was actually considering negative interest rates.  All these low rates have done is acted as a sop to feed he coffers of the stock market and investor class. Actually pumping the DOW, as well as NASDAQ, into the upper stratosphere – at the expense of the regular savers still trying to get 0.1% yield on bank savings accounts.  (The interest earned on our last Wells Fargo statement for the last quarter was a whopping 8 cents. And that is based not on a 'chicken feed' amount of $.)


"Well, why doncha get in the stock market then, dummy?"  


No, and we shouldn't have to! That is, risk our life savings on the volatility and vagaries of the markets.  Nor should we have to chase yield in other risky investments just to try to keep up, nor should any other American.  The whole issue now is that our economy is out of whack because too much has been surrendered to the speculators.  In addition, these twerps have accumulated a lot of their gains based on debt, or the use of leverage and excessive borrowing.    This was all highlighted in a Financial Times piece from April, 2018 ( 'IMF Sounds Alarm On Excessive Global Borrowing') where we read:


"The world's $164 trillion debt pile is bigger than at the height of the financial crisis a decade ago, the IMF has warned, sounding the alarm on excessive global borrowing.  The fund said the private and public sectors urgently need to cut debt levels to improve the resilience of the global economy, and provide greater firefighting ability it things go wrong."


So the warning alarms were being sounded about the markets' irrational exuberance even then.  But who was listening? Not Powell and the Fed, who kept happily going on with the near zero interest rate environment which was like putting Wall Street on crack.  


Thankfully, Jay Powell and the Fed seem to have come to their senses and grasp that some powerful medicine is needed to get current inflation under control (The Fed and Inflation  (WSJ p. A1). Fed Chair Jay Powell, in a recent confab in Jackson Hole, WY,  cited the example of former Fed chairman Paul Volcker, who drove the economy into a deep hole in the early 1980s with punishing rate increases to break the back of double-digit price gains.  In Powell's words,  invoking the title of Mr. Volcker’s 2018 autobiography, “Keeping At It.:


 “We must keep at it until the job is done,” 


This begs the question of how much of a short-term interest rate hike should be formalized today, at the Fed's meeting.  Let's grasp that the Fed has two mandates: full employment and stable prices.  Officials define the latter with a formal 2% inflation target. For most of the past two decades the Fed could focus on full employment because inflation seldom deviated much from 2%. Today, though, with inflation at 8.3%,  that deviation is past 6 percent, meaning the Fed has lots of ground to make up.  Even with the Fed's recent rate hikes, however, it is still trailing the inflation rate.  This isn't good enough and is why I believe the Fed needs to raise rates by a full one percent, not 3/4 % as the pundits suggest.


There is logic and reason behind this, despite the fact nervous Nellies will fear a recession.  That is, the Fed postponed proportionate response for so long that it now needs to make up for the lost time (and lost interest hikes)  or see inflation soar past 9%.  I realize people are already squawking about borrowing costs and their rising mortgages but what's worse, paying a bit more on the monthly mortgage or having your money become near worthless thanks to soaring inflation?  


Besides, someone has to feel the pain and now it's the borrowers' turn. We at the low end of the interest and savings curve felt it for years, as we watched the DOW soar to over 30,000 thanks to cheap money spurred on by near zero rates  and our own saving stagnate.  The markets profiting handsomely at our (savers) expense.


If Jay Powell is going to emulate Paul Volcker then he needs to act the way Volcker would in this same situation.  Volcker would raise the rate by 1% not only today, but have two more successive 1 percent rate hikes before year's end.  But how serious is Jerome Powell?  According to the front page WSJ piece, "Powell has stopped talking about a so-called soft landing, in which the Fed slows growth enough to bring down inflation without causing a recession. Instead, he has framed the Fed’s objective of bringing down high prices as an “unconditional".


Fed governor Christopher Waller meanwhile, informs us (ibid.):


If unemployment were to stay under, say, 5%, I think we could really be aggressive on inflation."


Then given current unemployment is under 5 % - NOW - this is the time to raise rates aggressively to try to catch up to inflation.  It is absolutely not the time to snooze or be excessively at the mercy of the 3/4 point pundits!


Economists basically expect Fed officials to forecast that their key rate could go as high as 4% by the end of this year.  This is not enough given how much havoc inflation is still wreaking.  The end target for the year  ought to be at least 5 percent. Forget "soft landings", there are too many Americans still suffering hard ones right now.


See Also:

by Robert Reich | September 22, 2022 - 7:09am | permalink

Basic Differential Equations & How They're Applied To Physics

 Having examined advanced methods of solving differential equations, including partial differential equations, e.g.

Solving Partial Differential Equations: Three Easy Examples & One (Somewhat) Hard 

it's useful to go back to basic DEs for the benefit of those who may either: a) want to review basic approaches or b) those who want to learn them for the first time.  This is in preparation to showing specific applications to physcs. Let's begin with the simplest first order equation imaginable, which is also variables separable:


 dy =  x dx

As with all differential equations, the solution is accomplished via the process called integration.  If we integrate both sides, we obtain:

ò dy =  ò x dx

Yielding:

y = x 2 / 2 + c

where c is some undefined (as yet) constant of integration. We call the above the "general solution" to the differential equation.  This general solution is, in fact, a family of parabolas:

                          General solution - family of parabola - for dy = x dx.

If we wanted to obtain the particular solution, we'd have to assign boundary conditions at the outset. Usually these designate what values x, y are to have at a particular point, and also often the first derivative (y' or dy/dx) at the same point.  

Kinematics-Dynamics:

To fix ideas in terms of an actual application we consider an example from kinematics.   This is the basic differential equation for rate of displacement:

d x/ dt =   v       or:   ò dx = v òdt 

Integrating:

òxo x dx = v òt dt

Let position:  xo  = 0  at time t = 0, 

which is effectively the boundary condition.  Now, v = at, so:

x  -   xo    =   òt dt  =  a òt t dt

=   ½ a t 2   +    C  

But:   C  = xo    so the general solution becomes

x  =   ½ a t 2  +    xo

And the particular solution  (since xo = 0) is:

x   =   ½ a t 2 

Another basic differential equation arises from Newton's 2nd law applied to the stretching force (F= - kx) on a spring, so that one may write:

F  =  ma =  m(d2x/dt 2) = - kx 

Or, isolating the 2nd derivative:

(d2x/dt 2)  +  (k/ m)  x  = 0

 d2x/dt 2  +  w 2  x  = 0

Where: w =  Ö(k/ m)  is  angular frequency 

And k is the spring constant.   This type of DE will have boundary conditions automatically set by the specific problem data or inputs, i.e. for mass, for k, for  w,  but sometime two variables may be given and the other(s) need to be obtained.

Capacitor Discharge:

Whenever a capacitor C is discharged through a resistor R. If the p.d. across C is V, then we have: Q = CV. The current discharged is then expressed:

I = - dQ/ dt

(i.e. Q, the charge, is decreasing as the current I increases)

 We know that Q = CV but by Ohm’s law: V = IR

 So: Q  = -CR (dQ/dt)

On separating variables we can integrate:

  ò0 t  dt/ CR  =  - òQo Q  dQ/ Q

Where:

òQo Q  dQ/ Q =  òQ Qo  dQ/ Q = ln (Qo/Q)

So that: 

t/  CR  = ln (Qo/Q)   or:

 e -(t/CR) =   Q/ Qo

Finally:

Q =  Qo  e -(t/CR)

Radioactive Decay:   

This bears an analogous form to capacitor discharge.

We define the activity of a radioactive source as:

A = dN/dt = - lN

Where  l  is the decay constant. The negative sign appended to the equation indicates that the amount N is decreasing with time t.  Separate variables:

dN/N  =  -  dt 

We can write the integral:

 òNo  dN/ N = ln (No/N) =  l  òt  dt   


The radioactive decay, then, is based on some original number of atoms No decaying with an activity l over time t:

N =  No  exp (-lt)

The half life is that time - call it  t = T ½   - for which half of the original atoms have disintegrated, i.e. No  ®  No  /2

Therefore:  No  /2  =  No  exp (-l T ½)

After dividing No into both sides and taking natural logarithms we get:

l T½  =  ln 2 = 0.693

 Or:  T ½ ln 2/ l  =   0.693/ l

 Using this format,  any sample or fossil with even a minuscule amount of radioactive material can be dated.  All we need know is that over the period defined as T½  half of the number of the remaining atoms decay and the activity A is in Becquerels (Bq).  

Thus, if  T½ = 15,000 yrs.  for  l = 200 Bq then if l  = 50 Bq now the sample is 45,000 years old.  A graphical depiction of generic radionuclide decay is shown below. 


                Radioactive Decay for a radionuclide with time T in millions of years (mY)

Here the vertical axis shows a relative scale for the amount or mass  of some, unnamed decaying isotope  which commences decay at some initial specified value, e.g. 1 gram then decreases to half that original amount in one half life – and thereafter to half that amount and so forth each additional half life.

Projectile motion:   A series of separate differential equations can be set up using the diagram shown below, based on projectile motion - incorporating gravity but neglecting air drag effects.

Diagram depicting projectile motion in 2 dimensions.

Here we can write the DEs for time t=0, at position x=0, y = 0:

v x =  dx/dt =  vo cos(q)

vy =  dy/dt  =  vo sin(q)

The force components at time t are:

F x =   0   and:   F y =   - mg

With these incorporated the differential equations become:

i) m d2x/dt 2  =   0   and ii) m d2y/ dt 2  =  - mg

Each equation above requires two integrations, yielding 4 constant of integration in all. 

For eqn. (i) we get on integrating:  dx/ dt = c1   Then:

ò dx  =   c1 òdt

Yields:  x = c1t + c2

Here for eqn. (ii) we have:

d2y/ dt 2  =  - g     So:   dy/dt = - gt  + c3

And:  y =  - ½ g t 2   +  c3t  +  c4

Based on the initial conditions given we have:

c1 =  vo cos(q),    c2 = 0,   c3 =   vo sin(q),  c4 =  0

Then the position of the projectile  at time t seconds after firing can be found from:  

x   vo cos(q) t    and:   y = ½ g t 2   + vo sin(q) t

Clearly, the projectile attains maximum altitude when its y -component of velocity is 0, i.e.

dy/ dt = 0 =  -gt  +  vo sin(q)

This must occur at time:

t' = vo sin(q)/  g

Then the maximum altitude is:

max =  ½ g t' 2   + vo sin(q) t' =

½ g [vo sin(q)/  g] 2  +  vo sin(q) [vo sin(q)/  g]

=  vo sin(q/  2 g


Suggested Problems:

1) Find the general solution to the equation:  dy/ dx = -x/y

2) Find the general solution to the equation: dy/dx = e x - y

3) The linearized equation for a simple pendulum with small oscillations is given by:

dq/ dt 2  =  -  q (g/ )  

Find the period and frequency

4) A capacitor of C = 500 mF is initially charged to a potential difference of 10.0 V and connected to a resistor R = 100 kW and allowed too discharge. Find:

a)     The initial charge on the capacitor

b)    The time constant for discharge through the resistor R.

c)    The time for the initial charge to decay to one half its initial value.

d)   The energy stored in the capacitor by this time.

5)A radionuclide sample of N = 1015 atoms undergoes decay at the constant average rate of dN/dt =  6.00 x  1011  /s.  From this information, find:

a) The Activity A

b) The decay constant l

c) The half- life  (½of the sample in minutes.

6) Using the example for the projectile motion, along with the accompanying diagram, find:

i) The maximum height attained when the launch angle = 30 degrees.

ii) Write an equation for the slope (dy/dx) of the path of the projectile at any point.

iii) Write the condition to obtain the range  R (maximum horizontal distance) of the projectile and find it.

iv) Write the equation for the slope of the path (e.g. dy/dx) of the projectile at any point.

v) From (iv) what can be inferred about the angle F ?