Using Differential Equations To Solve Missile Trajectory Problems

  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 =  v_o cos(q)

v_y =  dy/dt  =  v_o 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 =  v_o cos(q),    c2 = 0,   c3 =   v_o sin(q),  c4 =  0

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

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

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

dy/ dt = 0 =  -gt  +  v_o sin(q)

This must occur at time:

t' = v_o sin(q)/  g

Then the maximum altitude is:

y _max =  - ½ g t' ²   + v_o sin(q) t' =

- ½ g [v_o sin(q)/  g] ²  +  v_o sin(q) [v_o sin(q)/  g]

=  v_o sin(q) ² /  2 g

Example Problem:

Find:

i) The maximum height attained when the launch angle for a projectile is q = 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) From (iv) what can be inferred about the angle F ?

Solutions:

i) The maximum altitude is:

y _max =  - ½ g t' ²   + v_o sin(q) t' =

- ½ g [v_o sin(q)/  g] ²  +  v_o sin(q) [v_o sin(q)/  g]

=  v_o sin(q) ² /  2 g    

q = 30 degrees:

=  v_o sin(30) ² /  2 g   

But sin (30) = 1/2 so:

y _max =  v_o (1/2) ² /  2 g   

=   v_o (1/4) /  2 g   =  v_o /  8 g

ii) Slope at any point: 

dy/dx =  (dy/dt)/ (dx/dt) =

 - ½ g t ² + v_o sin(q)/v_o cos(q)

iii) Range (R) occurs when y = 0  

 This occurs at double the time for max. altitude since trajectory is symmetric.

So: 2t'  =  2 ( v_o sin(q) / g ) 

Then x = R  =  v_o cos(q) 2 ( v_o sin(q) / g ) 

=  ( v_o ² / g) sin(2q)

iv)  The slope dy/dx is (expanded from (ii)):

dy/dx = -2 v_o sin(q) + v_o sin(q)/ v_o cos(q) 

= -  [sin(q)/ cos(q)] =   -  tan (q)

This shows the angle F =   p   -  q

i.e. projectile returns to Earth at same angle it left.

Suggested Problem:

Find the  x- and y-coordinates of the points on the trajectory of a missile launched at an angle of 80 degrees with an initial velocity of 100,000 f/s if the air resistance is 0.01mv. Find the value of x and y after 10 seconds.

Data:    q = 80^o,   k = 0.01mv   v = 10⁵ fs^-1 and t = 10s

Earth's New 'Mini-Moon' 2024 PT5 Is Actually A Small Asteroid -- But Don't Worry About It Getting Loose

 

                                Artist's depiction of "mini-moon"  2024 PT5

The new mini moon, 2024 PT5, was first spotted in August by astronomers at Complutense University of Madrid using a powerful telescope located in Sutherland, South Africa. But since last Sunday it will be temporarily trapped by our Earth's gravity and orbit the globe - but only for about two months.

 In fact, this interloper is actually an asteroid about the size of a school bus at 33 feet (10 meters). As for posing any danger in striking our planet, no not at all. Even if it did somehow break loose and impact our planet, it would still have much less than the explosive Chelyabinsk event in February, 2023.  That small asteroid was at least 66 feet in diameter by way of comparison.

In any case, these short-lived mini moons are likely more common than we realize. For example, the 2022 NX1 asteroid joined Earth's orbit in 1981 and 2022, according to BBC News.  According to one astronomer familiar with the objects:

"This happens with some frequency, but we rarely see them because they're very small and very hard to detect. Only recently has our survey capability reached the point of spotting them routinely."

The discovery by Carlos de la Fuente Marcos and Raúl de la Fuente Marcos was published in The Research Notes of The American Astronomical Society:

A Two-month Mini-moon: 2024 PT5 Captured by Earth from September to November - IOPscience

Lead author Carlos de la Fuente Marcos noted in an email:

"This one won't be visible to the naked eye or through amateur telescopes, but it can be observed with relatively large, research-grade telescopes," 

It takes about 27 days for the moon to make one complete orbit around the earth. The asteroid disguised as a mini-moon will travel with us for nearly 57 days, completing almost one full orbit of Earth. After that, the rock will break free from our gravitational grasp on November 25, returning to its path around the Sun. It's expected to pass by again in 2055.

It might sound extraordinary to get a second moon, but according to NASA it’s quite common for asteroids and other space rocks to enter the gravitational pull of the earth. 

"Why didn't anyone know about this before? Like in the 19th century?"

Well, because the objects, like PT5 at 33' diameter, were simply too tiny to be picked up by the telescope of the time.

And so, unlike the moon, the asteroid 2024 PT5 won't be visible to the casual stargazer because of its size. It’ll be too small and too dim to be seen without a powerful telescope.

Want a moonlight dinner? Stick to when our own original Moon shines bright like at next Full Moon. In the meantime, rest easy as there's almost zero chance of getting loose and whacking our planetary abode.

See Also:

The Earth Now Has Two Moons—Here's How to See Them (sunset.com)

And:

Good News: Close encounter with a second moon (nbcnews.com)

 

Tim Walz Does Creditably In Fending Off Lie Tsunami of JD Vance in VEEP Debate

 

             Tim Walz has his hands full countering Vance's lies last night

"What is wrong with Trump’s running mate, JD Vance? Does he think Americans are so stupid that they’ll believe him when he says, with Ivy League smoothness and a practiced tone of faux sincerity, that up is down?

“It’s really rich for Democratic leaders to say that Donald Trump is a unique threat to democracy when he peacefully gave over power on Jan. 20,” 

The senator from Ohio claimed during the vice-presidential debate Tuesday night, earning himself a chapter in the annals of disingenuousness.-  Eugene Robinson, Washington Post

WaPo columnist Dana Milbank summarized last night's Veep debate perhaps most succinctly and accurately:

"It was a lie on top of another lie, supplemented by a pair of other lies, in support of an even bigger lie.  That was JD Vance’s modus operandi through 88 minutes."

For example, when Vance whined to moderator Margaret Brennan:

“Margaret, the rules were that you guys weren’t going to fact check!”

Bwahhaaa! In fact there were no such ‘rules’ it was a media assumption. 

That was after she pulled him up for one especially egregious whopper e.g.

 “You’ve got schools (in Springfield) that are overwhelmed, you’ve got hospitals that are overwhelmed, you have got housing that is totally unaffordable because we brought in millions of illegal immigrants.”

 All of which is utter bullshit, leaving out the point that Springfield arranged to have the Haitians move there to take work-jobs desperately needed but that there weren't enough locals to fill.

The Haitian immigrants in Springfield, as CBS moderator Margaret Brennan noted, have legal status. Their arrival in town, local residents and leaders have said, has helped revive the town, which lost a quarter of its population since the 1960s.

What Vance's petulant whine to Brennan disclosed is his fear of having his cavalcade of planned lies exposed by a moderator. When he thought he'd have free reign to run roughshod over telling the truth for nearly 90 minutes.

But swatting down JD’s stream of lies was almost as daunting as when Biden faced Trump’s barrage of baloney and BS back on June 27th.  Worse, Vance had a practiced debater's way of delivering them, with some polish and panache - mixed with lingo -- that sufficed to 'baffle with bullshit' many viewers.  Including some columnists at The Baltimore Sun who actually got snookered by Vance into believing he 'won'.   Had they paid attention to the response he gave about January 6, and whether he actually believed Trump won the 2020 election, they'd have seen it was straight liar's poker. The inability to acknowledge the truth that Trump lost and that his renegades were ready to hang Mike Pence in the insurrection, cost JD whatever meager creds he had up until then, and the debate.

Other wanton lies included that Trump had actually sought to preserve Obamacare and refine it, when in fact his endless chant was "repeal and replace".  Fortunately, Tim Walz came back hard on that one to clarify especially for those voters who may have short memories and be unable to separate fact from fiction.  Or were taken in by Vance's smooth delivery and snake oil salesman's patter.

Then there were Vance's incessant references to Kamala Harris did this, and Kamala Harris did that.  Like saying "Americans had their lives destroyed by Kamala Harris’s open border”.  Newsflash there is no Kamala Harris open border because she doesn't run the border, and isn't president. She was appointed Biden's "Border Czar"  but that wasn't to control the border but to go on a fact -finding mission to Guatemala, Salvador etc. to find out what was driving the mass migration of so many.

 Fortunately, Walz accused Vance of trying to “dehumanize and villainize other human beings” when he made false claims about Springfield, Ohio. But he also should have slammed Vance's mouth shut on tagging Kamala Harris with every kind of permutation or policy decision when such claimed ‘policies’ either never existed or did exist but she had no control of formulating in the first place. Since she is not the POTUS. she doesn't have the power to exercise any such decisions. 

Other Vance-bytes:

Vance said that Donald Trump supported states making their own abortion laws. The imp claimed that Trump said “the proper way to handle this … is to let voters make these decisions, let the individual states make their abortion policy,” 

That’s not quite right: Donald Trump declined to say whether he would sign a national abortion ban during the last debate with Kamala Harris.  He knew damned well he'd lose more female votes if he admitted that, just like Vnce knew if he came clean about it. So he lied.

Vance also denied that a Trump administration would create a federal pregnancy monitoring agency. Actually, it's a menstrual cycle monitoring policy which would come into effect if voters are dumb enough to vote Trump in again, and he implements his Project 2025 fascist dreamscape.  See:

Republicans | Women's Rights

The Menstrual Police are Coming: GOP's Plan for Total Control Over Women’s Bodies

by Thom Hartmann | October 1, 2024 - 5:57am | permalink

— from The Hartmann Report

“Menstrual police?” you may be thinking. That’s not possible! Not in America!

Read on.

The Comstock Act is an 1873 law that, if enforced, would outlaw all abortions in America by banning the shipping via mail, UPS, FedEx, etc., of any device, drug, or instrument that can be used to produce an abortion. It would even shut down hospital abortions.

It could also be used to empower new state or even federal police agencies specifically overseeing women violating its provisions. Like the menstrual police, which a Trump senior advisor just said was a very real possibility.

First the background.

» article continues...

Vance also said that he and Donald Trump were endeavoring to be “pro-family in the fullest sense of the word”, that he supports fertility treatments  (like IVF) and that he wants mothers to be able to afford to have babies.  But when Trump was pinned down on the IVF issue and asked to specify a policy he waffled and veered into the disconnected realm of taxes on tips. See e.g.

Jon Stewart: Trump Is The Opposite of Who His Supporters Claim He Is | The Daily Show (youtube.com)

Vance was also asked about his previous attacks on Donald Trump, including saying that he could be “America’s Hitler”. And here he was just as disingenuous to try to show it wasn't really him, but the media which put those errant thoughts into his brain.

Vance said he has disagreed with Trump but that he was wrong about the former president being another Hitler because he “listened to too many in the media” about him.  Adding;

When you screw up, when you misspeak, when you get something wrong and you change your mind, you ought to be honest with the American people.

ROTFL!  So let me get this straight: You have zero control over formulating your own thoughts, opinions and decisions.  Can we say pathetic? 

 No wonder Trump made you his VEEP, you're a bigger doormat than even Mike Pence was.

Another great shot by Tim Walz was when he mocked Vance’s statement about how Trump’s economic plans are being attacked by people with PhDs. I.e.:

"Economists, don’t be trusted. Science can’t be trusted. National security folks can’t be trusted. Look, if you’re going to be president, you don’t have all the answers. Donald Trump believes he does. My pro tip of the day is this: if you need heart surgery, listen to the people at the Mayo Clinic in Rochester, Minnesota, not Donald Trump. And the same thing goes with this."

To refresh memories, and as reported in The Wall Street Journal  ('Economists See More Inflation Under Trump', p. A2,  July 13-14):

 "Of the 50 who answered questions about Trump and Biden, 56% said inflation would be higher under another Trump term than a Biden term versus 16% who said the opposite."  

 The reaction of the professional economists were understandable given Trump is proposing to implement a "universal tariff" of 10%- 20%  on every country in the world. Contrary to Dotard's "economics" or fringe beliefs, tariffs aren't paid by foreign governments.  So he isn't hurting them.  

They are paid initially by U.S. companies that import whatever goods - whether baby formula or HDTVs - and then these are passed on to American consumers. Result? Higher cost, more inflation. Thus, Trump's tariffs would push costs up on just about everything and increase inflation dramatically.

Regrettably, as one MSNBC analyst noted, too many people - viewers will be taken in by Vance's slick delivery  (though laden with falsehoods) and not be able to recognize the content for what it is, BS. On the positive side, VP debates hardly ever matter in the scheme of elections. 

But the main takeaway from this one should be that JD Vance is capable of lying with a straight face, so no one should take anything he says with more than a grain of salt.  No matter how many pundits give the freak a 'W' based on style points.  

As for Kamala's ask for a 2nd debate with Trump, as MSNBC analyst Michael Steele put it, 'forget about it'.  Vance succeeded in repainting Trump for the low IQ, low information voters as a moderate 'winner', and reconstructed Reep prez candidate. So any more debates - with either Trump or Vance - would amount to a lose-lose.

  If Kamala wants to project a further voice out there and detail her plans, the best route is doing a Town Hall or two, which she did in 2019.

See Also:

In the News

The Dangerous Art of 'Sane-Washing': Lessons From the Walz-Vance Debate

by Peter Bloom | October 4, 2024 - 6:58am | permalink

In the wake of the recent vice presidential debate between Minnesota Gov. Tim Walz and Ohio Sen. JD Vance, political commentators have been abuzz with praise for Vance's performance. Many have both lauded and critiqued his ability to "sane wash" the extremist positions of his running mate, former U.S. President Donald Trump, presenting them in a more palatable, even respectable light. This phenomenon, while concerning in its own right, reveals a deeper and more insidious problem within our political discourse—one that extends far beyond the bounds of the Republican ticket.

The Illusion of Moderation

JD Vance, the
bestselling author turned venture capitalist turned politician, took to the debate stage with a clear mission: to repackage the Trump agenda in a way that would appeal to a broader audience. Gone were the inflammatory rhetoric and bombastic declarations that have become Trump's hallmark. In their place, Vance offered measured tones, appeals to compassion, and a veneer of reasonableness that seemed designed to make even the most controversial policies sound sensible.

» article continues...

And:

MSNBC's Nicolle Wallace claims worst moment for Vance was when he tried to 'mansplain' over muted mics | Watch

And:

Vance’s debate performance was a breathtaking exercise in sane-washing

And:

JD Vance | Tim Walz

Tim Walz Wins the Nice Guy Debate

by Heather Digby Parton | October 3, 2024 - 6:24am | permalink

— from Salon

Writing about a debate on the morning after always feels more like theater criticism than political analysis. How did they look, how did they sound, did they come off as authentic and real or were they phony and glib? Were they believable to the faceless Real Americans watching being asked to decide which of them to vote for? But that's what these televised debates really are. The substance is usually secondary because they've practiced their lines and have a specific message they want to impart regardless of the topic they're being asked to address. They're political rituals that we use to decide if the person appears to be someone we want to watch perform the role of whatever office they are seeking.

The worst debate ritual we've all ever witnessed happened last June when President Joe Biden was seen to be doddering and incompetent. It wasn't that most Democrats disagreed with his policies to the extent that he articulated them or were unhappy with his record, quite the opposite. It was his performance, and it resulted in him having to withdraw from the race. One of the best debates of the last few decades was the one after that, when Vice President Kamala Harris wiped the floor with Donald Trump. The former president's performance revealed him as unprepared and incompetent while the vice president was effective and commanding. Trump has retreated into a negative feedback loop ever since.

» article continues...

And:

The dangerous brilliance of JD Vance

And:

JD Vance

JD Vance Isn’t Just Thin-Skinned — He Thinks Whining Is a Winning Political Strategy

by Amanda Marcotte | October 2, 2024 - 6:06am | permalink

— from Salon

To know Sen. JD Vance is to dislike him. The Ohio Republican was relatively unknown to most Americans before Donald Trump picked him as a running mate, and few, if any, politicians have garnered such a negative reaction in such a short period. It's not just his rants about "childless cat ladies." Focus groups show that voters are well aware that, while Vance publicly praises Trump like he's a god, he talks smack about his boss behind his back. But I suspect, like Heather "Digby" Parton wrote at Salon Monday, "his nasty, cold personality" is a factor in Vance's unpopularity.

Vance can't seem to speak without whining. Every interview with him is a grievance-fest where he plays the victim of "the media," lies while falsely accusing his opponents of lying, and acts put out by inconsequential nonsense. He's as full of self-pity as Trump. Vance can be even more aggravating because, by all accounts, has a great life well beyond what he deserves: a beautiful family he doesn't appear to appreciate, a Senate seat purchased for him by a tech billionaire, and millions of dollars, despite not offering any real value to society or the economy.

» article continues...

 And:

At debate, Vance whines: You weren’t supposed to fact-check me!

 And:

Who won the debate? We asked swing-state voters.

Barbados PM Mia Mottley Schools Bibi Netanyahu on International Law, Morality and Temperance In Fiery U.N. Address Friday

 

                         Bibi holds up his flash cards during bellicose U.N. spiel

                           Bim's Mia Amor Mottley schools Bibi after his spiel

The  defiant Israeli autocrat Bibi Netanyahu used his speech to the UN General Assembly Friday to drill home his refrain that Iran is the "global ground zero of evil", and that Israel’s opponents are "anti-Semites" - oh, and that Hamas and Hezbollah will be "shown no quarter."

Thousands of protesters demonstrated against Netanyahu  in Manhattan on Thursday and Friday, and he addressed an almost empty Assembly Hall after many delegations walked out as he took the rostrum.  After all, why give time to a confirmed, unrepentant autocrat who is only using his war B.S. to further prop up his regime? Especially when this freak was responsible for the lapse in security that enabled the Hamas Oct. 7, 2023 attack in the first place?  

So now Bibi and his rogue military have left nearly 40,000 dead in Gaza, nearly two- thirds kids and women. That is along with reducing the narrow strip to uninhabitable rubble and disease. Then he had the West Bank blasted to bits before dispatching his air force to hit more than 2,000 targets last week  in Beirut leaving more rubble along with more than 600 people killed and roughly 2,000 injured. (WSJ, Sept. 28, p. A8)

But while the UN Assembly Hall was basically empty the public gallery was packed with Bibi's groupies and cheerleaders.  In his off the walls harangue, the Israeli dictator shouted:

“We face savage enemies who seek our annihilation. And we must defend ourselves against these savage murderers,” 

But Bibi wasn't telling a soul that among the Middle Eastern states Israel alone has nuclear weapons. At last estimate, over 200 of them - enough to level Iran, Iraq, Syria, Saudi Arabia and Lebanon to radioactive ash. The Israelis have had such capacity since the 1960s when PM David Ben - Gurion launched his nuclear reactor at Dimona much to JFK's chagrin. See e.g.

Kennedy, Dimona and the Nuclear Proliferation Problem: 1961-1962 | Wilson Center

 Then Bibi in his UN tirade had the chutzpah to yelp about opposing Iran and blocking it from obtaining nuclear weapons.

Meanwhile, Netanyahu made no reference to efforts by the U.S. and France to broker a 21-day ceasefire between Israel and Hezbollah in Lebanon. He also referred scornfully to the UN itself and the International Criminal Court prosecutor who applied for arrest warrants for Netanyahu, his defense minister Yoav Gallant and three senior Hamas leaders, accusing them of war crimes and crimes against humanity.

Fortunately, directly following Netanyahu’s fulsome diatribe, Barbados PM Mia Mottley ascended to the dais to bring some notes of badly needed sanity , declaring:

"We have seen the spread of the war from Gaza, to the consequences of the West Bank, to clearly what is now happening to Lebanon as we speak, with Israel. All of these have a legitimate concern as they rob the global community of financial oxygen and global resources.   As students of history we all know even the longest war came to an end. These wars will also come to an end but the questions are: When and at what cost? And with how much loss of life? Including how it will affect the children in the future and their every action for the next 50, 60, 70 years of their lives.

Innocent people are paying the price of the one thing that is theirs to give and they don't give it willingly. It is their life. Unless we address the root causes of these wars one by one, and the manners in which they are being sustained and financed, we will never know anything else then war and rumors of war. The transmission of these extreme scenes into people's rooms, bedrooms will trigger two extreme reactions, neither of which can be acceptable to us in the 3rd decade of the 21st century. We will either get the de-sensitizing of ordinary people to the loss of lives or the anger and inclination for vengeance that is spawned.

We need peace, and it must not be so hard to work for peace. And in the words of the Old Testament, much of which has guided peoples across this world, we see "an eye for an eye". But when we turn from the Old Testament to the New we see - as we do in Romans:

"Vengeance is mine sayeth the Lord."   Not any country. Not any human being.  So that the Bible can't be used as a convenient aid when it suits us and rejected when it doesn't. In the midst of this maelstrom we were very clear. My country took the step of recognizing and establishing diplomatic relations with the state of Palestine. And in spite of having supported a two state solution since 1969. We did this because we recognize the people of Palestine are entitled to recognition and full integration into and support from the international community.

We therefore share with other nations in congratulating the state of Palestine in taking its place among the nations in September this year.  And let me be clear, we condemn the actions of Hamas on October 7th.  But we equally and strongly condemn the humanitarian catastrophe in Gaza which is the result of the disproportionate use of force by Israel.   A two state solution, no matter how elusive it may appear to be now, is the only answer.  We have seen other battles in mankind's history in which we didn't think we could succeed but we did. We didn't think we could abolish slavery but we did. We removed apartheid in South Africa.

These battles are not beyond our creativity, our capacity, our resilience to stop them.

There is no justification for it. And that is why treaties exist governing the rules for engagement for war.  Because we as human beings know better, have learned better and committed to better.  And we are now threatened with the growing spread of chronic non-communicable diseases. We cannot afford the distraction of war. If ever there was a time to pause and reset it is now."

Mottley pointed out that “the anger and mistrust of our citizens in institutions, in leaders, and in multilateralism and its processes which exclude, while yielding much talk and little action, is very real.”

She called out the hypocrisy of the Security Council hierarchy, which “suggest(s) that some are full members and others are only part members, part-time, or occasional members,” which “has no place in the twenty-first century.”

Also needing a reset is trust in the international financial architecture. “Restricted access to capital,” Mottley said, “its proportionately high cost, its inadequate scale and the overwhelming burden of debt, are now combining to force governments in the world’s poorest countries and frankly, across many vulnerable middle income countries, to devote more resources to debt service than to health, education, and infrastructure combined.”

As Mia had five times the audience Bibi did, one hopes he will pay attention to her warnings on reckless assaults on innocents, and disproportionate military response and aggression , especially in causing such wanton destruction in Gaza and Lebanon. To see her full UN speech go to this link:

Barbados PM’s extraordinary attack on Netanyahu for selective use of Bible in UN | Janta Ka Reporter - YouTube

See Also:

Gaza Is A Humanitarian Catastrophe: Citizens There Should Be Offered An "Escape Hatch" - As Refugees 

And:

• Other Voices Weigh In On The Unfolding Gaza Humanitarian Disaster...