Ancora Imparo

14 November 2015

Motion Along a Curve, Part 3

Filed under: mathematics,physics — Darmok @ 15:58 UTC
Tags: , , ,

See Part 1 and Part 2, where I discuss trying to find equations of motion for a ball rolling along a track defined by an arbitrary function y(x). So far, I’ve worked out my general approach, and tested it on the very simple case of an inclined line. I do want to tackle some complex functions, but first I want to summarize the method, and to incorporate some changes I learned as I tried my first solution.

I had initially stated I wanted to start with x(0) = 0, to keep it simple. But I didn’t end up needing this restriction. My equation for velocity used the initial yi, which we got from plugging in the initial value of x. Also, we used the x(0) = 0 to find the constant after integrating. But these happened later in the process.

I also had started with an initial velocity of zero. That did make a difference. But looking back on part 1, I don’t think it would complicate the equation too much, and if it’s zero it will be an extra term to just drop out. Let’s go back to the conservation of energy equation, and keep vi in this time.

\displaystyle U_i + K_i = U + K

\displaystyle mgy_i + \frac{1}{2}m{v_i}^2 = mgy + \frac{1}{2}mv^2

\displaystyle gy_i - gy = \frac{1}{2}(v^2 - {v_i}^2)

\displaystyle 2g(y_i - y) = v^2 - {v_i}^2

\displaystyle 2g(y_i - y) + {v_i}^2 = v^2

\displaystyle v = \sqrt{2g(y_i-y)+{v_i}^2} \blacktriangleleft

I’m leaving the ± in this time. Strictly speaking, I’m not treating this as the magnitude of the vector, since magnitudes must be positive. Rather, I want to consider a velocity vector that can point ether forwards or backwards along the direction of the curve. I’m going to allow motion in both directions, not just forward.

Recall the graph showing components of the velocity vector:

Graph of arbitrary curve showing tangent line with right-triangle components

Graph generated in Python/Matplotlib.

Now I want to find the x-component. As I discovered last time, I don’t need to bother with the y-component — once I find an equation for x(t), I can use that directly to obtain y(t). The x-component will be

\displaystyle v_x = v \frac{1}{\sqrt{(y')^2+1}}

\displaystyle v_x = \pm \sqrt{\frac{2g(y_i - y) + {v_i}^2}{(y')^2+1}}

\displaystyle \frac{dx}{dt} = \pm \sqrt{\frac{2g(y_i - y) + {v_i}^2}{(y')^2+1}} \blacktriangleleft

Since I will have y and y′ in terms of x, I would need to rearrange to solve the differential equation. Let’s see how far I can take the general case:

\displaystyle \pm \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = dt

\displaystyle \pm \int \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = \int dt

\displaystyle \pm \int \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = t \blacktriangleleft

where I did not include a constant of integration on the right side, since it can be absorbed into the constant that the left integral will produce.

So, the general approach should be as follows: Given our equation y(x), find y′(x). Plug in those expressions, plug in the initial velocity, and plug in the initial height y[x(0)]. Integrate, and solve for x in terms of t to get x(t), then plug that into y(x) to get y(t). I’ll test if this approach can actually work in the next post.

31 October 2015

Motion Along a Curve, Part 1

Filed under: mathematics,physics — Darmok @ 02:24 UTC
Tags: , , ,

Earlier this week, I saw an intriguing problem suggested, one involving motion along a curve. The idea is this: in elementary mechanics, we learn about how an object will move when placed on an inclined surface. If we restrict motion to two dimensions, this surface can be represented by a line. With gravity as the propelling force, can we find a general approach to deriving its motion along an arbitrary curve? In other words, given a function f(x), can we find functions describing the position (and therefore velocity and acceleration) over time?

Here are my initial thoughts. One, I am sure that this type of problem has been analyzed before. That doesn’t matter; I am going to try figuring it out myself. Two, I think we can use conservation of energy to determine the object’s velocity at any point. However, that will depend on knowing its height, and I don’t know how this approach would handle the object flying up off a short hill, or dropping away from a cliff. So three, I am going to restrict the object’s motion to the curve — consider it a track, rather than a 1D surface. That means that four, I can use the derivative to find the direction, but that means the object can only move forward. Five, I have to ignore friction. My conservation of energy approach will require that all energy be potential or kinetic; I won’t know hot to deal with losses to friction. Six, I am using constant downward gravitation. Seven, I am starting with initial position x=0 and initial velocity of 0. I don’t believe this method requires it, but I am already concerned about the complexity of the math. And seven, I am planning to obtain a velocity equation and integrate to find position, but I don’t know if the equation will have an analytic solution (or if I will know how to integrate it).

These are some significant limitations, of course. If this approach works, I suspect that it could be expanded to deal with several of these. For instance, we could keep in terms for initial position and velocity. Gravity or whichever force could be represented by a vector field rather than a constant force; this would make the potential energy term much more complex. I also wonder if I could incorporate friction by adding in an extra term, but I think this would turn the equation into a more complex differential equation.

Let’s start with a simple case that we can solve using conventional means, so we’ll have an answer to check later. I can also use this to check my conservation of energy approach. Let’s have the object rolling down an incline of 30° (π/6), starting at a height of 1. The ball will roll down to the right (I find it more intuitive to imagine a rolling ball rather than a frictionless object sliding, especially if the surface is curved). The equation for this surface will be

\displaystyle y(x) = -\frac{x\sqrt{3}}{3}+1 \, \bullet

where the slope is -tan(π/6).

Graph of y=-sqrt(3)/3*x + 1

You can see that it would form a right triangle with height 1, length √3, and hypotenuse 2. The slope is therefore 1/√3, or √3/3. Perhaps it would have been better to use an incline of 60° (π/3) so that the slope would be -√3, but I like this one better. Let’s try the conservation of energy approach to see what the velocity would be when the ball reaches the bottom (that is, y = 0). The total energy E is the sum of the potential energy U and the kinetic energy K. This should remain constant, so

\displaystyle E_i = E_f

\displaystyle U_i + K_i = U_f + K_f

We know that potential energy is given by

\displaystyle U = mgh

assuming constant gravitation g (the acceleration due to gravity), with mass m and height h; and kinetic energy is given by

\displaystyle K = \frac{1}{2}mv^2

where v is the magnitude of the velocity vector v. I confess I am not very proficient with vectors, but clearly the ball is moving in two dimensions and we will need both components in the x and y directions. I am going to try to not be sloppy but to carefully think about what we mean by v or v every time I write it.

We assume that the initial velocity, and therefore kinetic energy, is zero.

\displaystyle U_i + K_i = U_f + K_f

\displaystyle mgh_i + \frac{1}{2}m\cdot0^2 = mgh_f + \frac{1}{2}mv_f^2

The height is y, and we can divide both sides by the nonzero mass m:

\displaystyle gy_i = gy_f + \frac{1}{2}v_f^2

Let’s solve for v:

\displaystyle 2g(y_i-y_f)=v_f^2

\displaystyle v_f = \sqrt{2g(y_i-y_f)} \blacktriangleleft

This should be the general case, so if this checks out, we can use this equation to develop the method. As a check, if the units of g are m/s² and y is in m, the units will be m/s, appropriate for velocity. Also note that this is simply the magnitude of the velocity vector (the speed); the actual vector could point in any direction if all we know is conservation of energy. That’s why I earlier constrained the ball to move along the curve — that will give us our direction. Plugging in our initial and final values of y, we get

\displaystyle v_f = \sqrt{2g(1-0)} = \sqrt{2g}

Next, to verify this solution, let’s solve using the traditional free-body approach. Gravity applies a force downwards of W=mg. This can be split into two components, one perpendicular to the surface mg cos(θ), and one along the surface mg sin(θ). The perpendicular force will be balanced out by the equal and opposite normal force, leaving a force of mg sin(θ). This is simple motion in one dimension, and let me introduce a variable s to represent its position along the surface (so I don’t generate confusion with our x’s and y’s). Think of it as placing a tape measure from the bottom of the incline up to the starting point. We know that for a given constant acceleration a,

\displaystyle v = \int \! a \, dt = at + v_0

\displaystyle s = \int \! v \, dt = \int \! (at+v_0) \, dt = \frac{1}{2}at^2 + v_0 t + s_0

By Newton’s second law, F = ma, so a = g sin(θ) and will be negative, since it’s pointing towards decreasing s (since this is motion in one dimension, we can use the sign to expression direction rather than need vectors). Initial velocity is zero, and initial position is 2 (the distance along the incline, if the bottom is at s=0).

\displaystyle 0 = -\frac{1}{2}g \sin(\frac{\pi}{6}) \cdot t^2 + 0t + 2

\displaystyle \frac{1}{2} g \left( \frac{1}{2}\right)t^2 = 2

\displaystyle \frac{gt^2}{4} = 2

\displaystyle t^2 = \frac{8}{g}

\displaystyle t = \pm \sqrt{\frac{8}{g}}

I can discard the negative solution, since I’m interested in the behavior starting from time = 0, not how it might have been launched before to come to a temporary stop at time = 0. Let’s plug this time to get our velocity from v = at (since the initial velocity is zero):

\displaystyle v = at = -g\left(\frac{1}{2}\right)\sqrt{\frac{8}{g}} = - \sqrt{\frac{g^2}{4}}\sqrt{\frac{8}{g}} = -\sqrt{2g}

This velocity is negative since it points downwards along the s direction, along the incline. Its magnitude will be the absolute value which is what we obtained earlier using the conservation of energy method. Armed with this success, and with some test values, we’ll be ready to work on the actual problem in the next post.

Edit: I added a graph.

8 January 2009

Individual Action is Not Enough

Filed under: environment,Uncategorized,video — Darmok @ 06:25 UTC
Tags: , ,

The Canadian chapter of the World Wildlife Federation produced this cool video/commercial:

Most people try, to at least some degree, to take steps to help the environment. And these small changes, when summed across the whole population, are significant. But still, the collective action of individuals can only do so much — government and industry need to be on board, too. Unfortunately, in the United States, leadership from the federal government has been lacking (and at times, actively impedes) so state and local governments and industry have had to take their own steps. There is some hope, though, that this will change when the Obama administration takes office (I hope to write more on this in a later post).

1 December 2008

Questions on How to Live a Green Lifestyle

Filed under: environment — Darmok @ 05:25 UTC

Most Americans are interested in living more environmentally friendly lifestyles, yet it’s not always easy to know which practices really are best. New Scientist takes a look at these “Dumb Eco-questions You Were Afraid to Ask”. Hope this helps clear up some doubts or misconceptions.

20 November 2008

President-elect Obama Delivers Strong Statement on Climate Change

As we prepare to move past President Bush’s disastrous environmental policies, I’ve been interested to see what President-elect Obama plans to do for the environment. The economy has garnered the most attention, and in the short term, is more important. But continued neglect of the environment will, in the long-term, lead to crises both in the economy and in other sectors.

President-elect Obama addressed the attendees of the Governor’s Global Climate Summit in a four-minute video (high-resolution version is available at; full text of speech at the end of this post).

He thanked the governors for their work (Governor Schwarzenegger of California along with governors of other U. S. states are hosting the Governor’s Global Climate Summit; leaders of key nations around the world are attending) and also thanked businesses for their efforts, going on to remark “But too often, Washington has failed to show the same kind of leadership. That will change when I take office. My presidency will mark a new chapter in America’s leadership on climate change that will strengthen our security and create millions of new jobs in the process.”

President-elect Obama went on to deliver more specific goals: “That will start with a federal cap-and-trade system. We’ll establish strong annual targets that set us on a course to reduce emissions to their 1990 levels by 2020 and reduce them an additional 80 percent by 2050. Further, we’ll invest $15 billion each year to catalyze private sector efforts to build a clean energy future”, indicating plans to invest in renewable resources as well as nuclear power and clean coal technology. He intends for this to help the economy as well, creating jobs and helping industry.

Mr. Obama also indicated a change in the way the U. S. has participated on the international stage, stating that the U. S. would work with and depend on other nations: “And once I take office, you can be sure that the United States will once again engage vigorously in these negotiations, and help lead the world toward a new era of global cooperation on climate change.”

Perhaps the most significant statement is the strong importance Mr. Obama still places on environmental problems, despite the problems with the economy. As John Broder writes in the New York Times, “State officials and environmental advocates were cheered that Mr. Obama choose to address climate change as only the second major policy area [after the economy] he has discussed as president-elect.” Reaction from environmental groups appears quite favorable.

The CEO of the World Wildlife Fund (WWF) praised President-elect Obama’s remarks: “Today President-elect Obama gave us his first official statements on climate and without a doubt he nailed it. He sees clearly the huge risk that climate change poses to our economy and our future, and he understands that solving climate change is a foundation for a global economic recovery.

Writing in the Sierra Club blog, Heather Moyer called the speech “very enjoyable”. And Peter Miller, in the National Resources Defense Council blog, wrote “Looking very presidential, Obama enunciated an unambiguous commitment to enacting a federal cap and trade program with tight annual caps leading to an 80% reduction in emissions by 2050.  The contrast with President Bush’s stance on climate change was abundantly evident to everyone.  It was the first time I’ve ever seen a standing ovation for a video.”

I look forward to more. Below is a transcript of the speech, taken from Grist with slight editing.

Let me begin by thanking the bipartisan group of U.S. governors who convened this meeting.

Few challenges facing America — and the world — are more urgent than combating climate change. The science is beyond dispute and the facts are clear. Sea levels are rising. Coastlines are shrinking. We’ve seen record drought, spreading famine, and storms that are growing stronger with each passing hurricane season.

Climate change and our dependence on foreign oil, if left unaddressed, will continue to weaken our economy and threaten our national security.

I know many of you are working to confront this challenge. In particular, I want to commend Governor Sebelius, Governor Doyle, Governor Crist, Governor Blagojevich and your host, Governor Schwarzenegger — all of you have shown true leadership in the fight to combat global warming. And we’ve also seen a number of businesses doing their part by investing in clean energy technologies. But too often, Washington has failed to show the same kind of leadership. That will change when I take office. My presidency will mark a new chapter in America’s leadership on climate change that will strengthen our security and create millions of new jobs in the process.

That will start with a federal cap-and-trade system. We’ll establish strong annual targets that set us on a course to reduce emissions to their 1990 levels by 2020 and reduce them an additional 80 percent by 2050. Further, we’ll invest $15 billion each year to catalyze private sector efforts to build a clean energy future. We’ll invest in solar power, wind power, and next generation biofuels. We’ll tap nuclear power, while making sure it’s safe. And we will develop clean coal technologies.

This investment will not only help us reduce our dependence on foreign oil, making the United States more secure. And it will not only help us bring about a clean energy future, saving the planet. It will also help us transform our industries and steer our country out of this economic crisis by generating five million new green jobs that pay well and can’t be outsourced.

But the truth is, the United States can’t meet this challenge alone. Solving this problem will require all of us working together. I understand that your meeting is being attended by government officials from over a dozen countries, including the U.K., Canada, Mexico, Brazil and Chile, Poland and Australia, India and Indonesia. And I look forward to working with all nations to meet this challenge in the coming years.

Let me also say a special word to the delegates from around the world who will gather at Poland next month: your work is vital to the planet. While I won’t be president at the time of your meeting and while the United States has only one president at a time, I’ve asked members of Congress who are attending the conference as observers to report back to me on what they learn there.

And once I take office, you can be sure that the United States will once again engage vigorously in these negotiations, and help lead the world toward a new era of global cooperation on climate change. Now is the time to confront this challenge once and for all. Delay is no longer an option. Denial is no longer an acceptable response. The stakes are too high. The consequences, too serious.

Stopping climate change won’t be easy. It won’t happen overnight. But I promise you this: When I am president, any governor who’s willing to promote clean energy will have a partner in the White House. Any company that’s willing to invest in clean energy will have an ally in Washington. And any nation that’s willing to join the cause of combating climate change will have an ally in the United States of America. Thank you.

2 September 2008

Sarah Palin’s Anti-Science and Anti-Environment Policies Are Worrisome

Filed under: environment,global warming,politics,science — Darmok @ 06:37 UTC
Tags: ,

Senator John McCain, the presumptive Republican presidential nominee, just announced Alaska governor Sarah Palin as his running mate. She was a surprise pick and is relatively unknown, but what I’ve found so far is somewhat disturbing. While I haven’t made my final electoral decision, what I do know is that I don’t want another George W. Bush.

Wired Science, part of the Wired blog network, discusses her views on teaching creationism in public school science classes. (Merriam-Webster defines “creationism” as “a doctrine or theory holding that matter, the various forms of life, and the world were created by God out of nothing and usually in the way described in Genesis [the first book of the Judeo-Christian Bible].”) They refer to an article in the Anchorage Daily News covering a 2006 Alaska gubernatorial debate:

The volatile issue of teaching creation science in public schools popped up in the Alaska governor’s race this week when Republican Sarah Palin said she thinks creationism should be taught alongside evolution in the state’s public classrooms.

Palin was answering a question from the moderator near the conclusion of Wednesday night’s televised debate on KAKM Channel 7 when she said, “Teach both. You know, don’t be afraid of information. Healthy debate is so important, and it’s so valuable in our schools. I am a proponent of teaching both.”

The article goes on to point out:

The Republican Party of Alaska platform says, in its section on education: “We support giving Creation Science equal representation with other theories of the origin of life. If evolution is taught, it should be presented as only a theory.”

This stance alone is a significant strike against her. However, her anti-environment policies are also troubling. For instance, she told NewsMax, “I’m not one though who would attribute [global warming] to being man-made.” As I discussed in a previous post, all major scientific societies concur that humans are responsible for climate change. Senator McCain, as well as Democratic nominee Senator Barack Obama and his running mate Senator Joe Biden, all agree that climate change is a real threat and have proposed plans to combat it.

It’s not surprising, therefore, that her policies appear to show general disregard for the environment, especially with regards to her strong advocacy for oil drilling. For instance, she stated, “I beg to disagree with any candidate who would say we can’t drill our way out of our problem…”, as quoted in Investor’s Business Daily (IBD) and “When I look every day, the big oil company’s building is right out there next to me, and it’s quite a reminder that we should have mutually beneficial relationships with the oil industry” as quoted in Roll Call. She supports opening up the Arctic National Wildlife Refuge (ANWR, commonly pronounced “AN-war”) for drilling, a move generally opposed by environmentalists as well as Congress. Expressing her frustation, she stated to IBD, “But these lands [ANWR] are locked up by Congress, and we are not allowed to drill to the degree America needs the development…”; to Lawrence Kudlow on CNBC, “Very, very disappointed in Congress though [for not voting on drilling in ANWR]”; and so on. Both Senators Obama and McCain opposing drilling in ANWR, and she has attacked Senator McCain for this stance: “I have not talked him into ANWR yet…I think we need McCain in that White House despite, still, the close-mindedness on ANWR” (Lawrence Kudlow, CNBC).

Nor has Alaska, under Mrs. Palin’s governorship, promoted environmental issues. In Massachussets v. Environmental Protection Agency, when twelve states as well as several cities and environmental organizations sued the EPA to regulate carbon dioxide and other greenhouse gases, Alaska argued against them. (In a split decision, the Supreme Court largely agreed with Massachussets et al; see my previous post.)

Earlier this year, the Interior Department listed the polar bear as a threatened species under the Endangered Species Act (ESA). Somewhat bizzarrely, Governor Palin claims that polar bears are not threatened (“In fact, the number of polar bears has risen dramatically over the past 30 years” she states). She opposed the ESA listing and Alaska now plans to sue the Interior Department. Similarly, Governor Palin is opposing plans to list beluga whales as endangered, as it could damage Alaska’s economy.

Eight years of disregard for science and for the environment is enough; I don’t think I want to see someone like this in high office, certainly not in a position where she could become president. If anyone has any examples of Governor Palin promoting science or the environment, please let me know.

29 May 2008

Incredible Photograph of Phoenix landing on Mars

Filed under: astronomy,science,visualization — Darmok @ 05:55 UTC
Tags: , , ,

Phoenix, a NASA robotic probe, landed successfully on Mars on May 25. It landed in the north polar region of Mars, at around the equivalent latitude of northern Alaska, and it will study Mars’ soil to look for clues of past water patterns and if it was ever hospitable to life.

Incredibly, as it parachuted down towards the surface, its picture was taken by a satellite orbiting Mars, the Mars Reconnaissance Orbiter (MRO)! From an amazing  distance of 750 kilometers (470 miles), it snapped this photograph of Phoenix parachuting towards Mars. This is the first time one probe has photographed another landing on a planet.

MRO photograph of Phoenix parachuting to Mars
See full-sized version. Credit: NASA/JPL-Caltech/University of Arizona

To see how this fits in to the landing, take a look at this cool animation of Phoenix landing, produced by MAAS Digital and NASA.

22 April 2008

Happy Earth Day!

Filed under: environment — Darmok @ 06:30 UTC

Earth Day flag
Commonly used Earth Day flag. Source: Wikipedia.

April 22 is celebrated as Earth Day in the United States. (In the rest of the world, the March equinox is chosen.) Please use this day to reflect on our planet, our relationship with it, and how our species can exist in harmony with other lifeforms. Help to ensure our children and their children will be able to enjoy our home.

Make every day Earth Day.

10 April 2008

Seafood Watch Helps Consumers Choose Sustainable Seafood

Filed under: environment — Darmok @ 04:46 UTC
Tags: , ,

An article in last month’s Scientific American, “Fishing Blues” highlights the problems that fishing poses for our marine life. As Earth’s population swells in both number and appetite, our fishing takes its toll through various harms, from overfishing to habitat destruction. And in addition to the inherent loss of losing biodiversity, this will have major impacts on humans — whether from simple shortages to far-reaching effects of damaged ecosystems.

Governmental regulations are important, but the most powerful force is that of the consumer. By choosing what to buy and what to avoid, consumers set the priorities for the industry. Clearly, eating sustainable vegetarian food in lieu of seafood or other animals is preferable, when possible. But for those times when one does wish to eat seafood, the Scientific American article points out a useful resource: Seafood Watch (, Wikipedia), a program run by the Monterey Bay Aquarium.

Screenshot of Seafood Watch

You can browse through different seafood or search for the one you want. For different areas of the U.S., they have regional guides categorizing common seafood into best choices, ones to select with caution, and ones to avoid. PDF pocket guides are available as well. (You can also access a streamlined mobile version at tricky aspect, though, is that the same fish can be sometimes be a good or bad choice, depending on where or how it was caught. For instance, U.S. mahi mahi is a good choice, but not necessarily from elsewhere in the world (due to U.S. policies regulating its fishing). This means that you will have to look at labels at markets or ask your server at restaurants to determine if a certain menu item is a responsible item or not. If your server doesn’t know, ask him or her to ask the chef, and if the origin still can’t be reliably determined, select something else. If people keep asking questions, perhaps next time they’ll make sure they know where their seafood comes from.

Seafood Watch also has a lot more information, including highlighting which fishing practices are harmful and why, and other actions you can take.

29 March 2008

Earth Hour is Today! Google Gets Involved, Too

Filed under: environment — Darmok @ 18:37 UTC
Tags: ,

Dark version of Google's home page for Earth Hour
Google changes its background color to black in observation of Earth Hour 2008.

Today is Earth Hour! If that time hasn’t already passed for you, please remember to turn off your lights from 8–9 p.m. today. And even if it has passed, please remember that, ideally, this should be part of an overall energy-conserving lifestyle. Plan for regular periods of very low energy use, and learn how much you can do even with turning some things off.

And in case you missed it, Google has redesigned their home page for today. They’ve changed their color scheme to use a dark background. In my memory, this is unprecedented. I have never seen them take up an initiative like this, and while I have seen them change their logo on numerous occasions, I do not recall them ever changing their whole color scheme like this. I am really impressed that they did this—they have the potential to reach so many people and what a great way to really call attention to Earth Hour. Of course, they have a long history of supporting environmental projects. They include a prominent link explaining their support of Earth Hour so anyone will be able to read about why they’re doing this.

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