In today’s inquiry course, we began our investigation about light:

At some point, we were discussing what it would look like if you were sitting in a dark room looking out into the hallway, and your friend was shining a flashlight down the hall. The major predictions that came up were these:

Idea #1: You’d see a “concentrated” beam of light. Some thought this would be like a narrow beam, others thought it would look bigger the farther it goes down the hall away from the flashlight (like a cone of light).

Idea #2: You’d see a beam of light, but that some of the light would “spill” or “leak” into the room you are in. That light  that spilled over would light up some of the room by the dorr, but they didn’t think it would be able to go around and light the whole room (like the corners).

Ideas #3: Everything in the hallway would just be “lit up” inside, allowing you to see what’s inside. Some said it would be brighter the closer you are to the flashlight (and dimmer farther away) in a long hallway, but that it would appear equally bright in a smaller hallway. Some with this idea talked about the “glow” of the light filling the space, while others talked about the light bouncing around off the walls to get everything lit up.

Idea #4: You’d mostly just see dust or particles shining in the air. Some thought you’d see dust only with a faint light (or light far from flashlight), because a strong light would just show the concentrated beam. Others thought that you would need a bright light in order to see the dust, because the light needs to reflect off the dust strongly.

 Idea #5: The side walls of the hallway would only would be dimly lit, but at very the end of the hallway there should a spot of light on the wall. If you were just looking in but not down the hallway, you’d see the dimly lit walls due to “ambient” light everywhere. But if you looked down the hall, you’d see the spot of light down at the end.

During and after observations, students noticed and wanted to talk about a variety of things, including

(1) The fact that they didn’t see a beam of light. This was the most common reaction, and several students talked about how they must have been thinking about like “cartoon” light.

(2) The fact that they did see some light get out of the hallway and into the room through the door, but it didn’t leak through, like they had expected. They talked about how the door angled the light in a particular way based on where the flashlight was located. On group mentioned how we only see the light out of the room because it hit the floor–we couldn’t see it traveling to the floor.

(3) They also saw that there was a specific starting place where the walls began to be lit, and the boundary of that lit places was curved. There were ideas about how the mirror in the flashlight causes this curviness. Other thought the circular shape of the flashlight hole was the cause. Some thought it might be both.

(4) There was also some discussion about whether the dark spaces were completely dark or just somewhat darker than the bright regions. One group explained that it didn’t seem completely dar (just much darker), and that maybe light was reflecting off the walls and getting back to those areas that don’t get direct light. This started to touch upon an idea that the reflected light might be dimmer than direct light.

(5) Some students noticed that there was a circular spot down at the end, and they used this as evidence that there must be a beam even though we can’t see it until it hits the wall. Other students said that they didn’t want to call such an “invisible beam” a beam, but something else instead. This conversation also started to touch upon ideas about what the mirror was doing. Some thought the mirror was “funneling” the light some how, making it more concentrated into a beam, while others thought the mirror was just making the light that would have gone backwards go forward.

Students also had various other questions, including :

(1) Why is the lit space curved the way it is?

(2) What would be different if we just used a regular bulb instead of a flashlight? Would different shaped bulbs make a difference?

(3) What would change if the back end of the hallway was a mirror? Would the dark spots behind the flashlight now be more lit?

(4) The shape of the light that made it out of the hallway was affected by the shape of the door. Would a circular hole make circular spot of light? Triangle hole triangular?

(5) What would change if we added lots of the dust? Would we see a beam then?

We have a lots of good starting ideas and questions, and there are lot of places that I want to work on getting them to clarify and be more specific about what they are thinking and why. I’m excited to see where we go, but I’ll have to wait a week.

There is an undergraduate student in our physics teaching program who wants to do his undergraduate thesis around using video analysis for physics labs. I am not supervising this research, but the student has come to me for some guidance.

This is how the student describes his work in an email to me:

The research I am doing involves using Logger Pro software to analyze motion videos for physics labs. The main goals of this research can be divided into three areas.

• Create 2-3 labs involving logger pro that can be used at the high school level, but with room for adjustments to be adapted for other levels.
• Design a questionnaire to be given to two classes; one that is using the logger pro software in their labs versus one that is undergoing the “traditional” physics instruction without logger pro.
• Create a comprehensive instructional guide for others to use logger pro in their own classrooms (for students and teachers).

For the most part, I feel like I can handle making the labs and the instructional guide, but I could use some guidance for the questionnaire. I also need some help in how I should put this together to make it look more like a legitimate research project than a mess of ideas put together.

Based on my experience working with graduate students on their Masters theses, I am not surprised to see new education researchers wanting to jump to developing, implementing, and testing curriculum. I am also not surprised that faculty–at least those who are not trained in education research–would encourage students to go this route. To many, it seems like the “science-y” thing to do–develop a product, run an experiment, take quantitative data, and compare outcomes.

My responsibility, as I see it, is to help new researchers do research that (1) contributes to the knowledge base on teaching and learning AND (2) helps them develop important skill for teaching. The balance between these two goals is dependent on the project and the student. What I don’t want them doing is re-inventing wheels and run wheel races. We don’t need more of that.

So the two things I would love to hear about from everyone is the following:

(1) What curriculum do you use or have you developed that for video analysis? And can you share it with me? What pedagogical philosophies or strategies encapsulate your use of video analysis? How does this fit within the bigger picture of your course?

(2) From the practitioner side, what questions, concerns, and issues do you have about video analysis? What questions could an education researcher pursue that would contribute to your practice or to your understanding of the teaching and learning you do with video analysis?

Today, I visited a high school physics class of a brand new teacher.  Here’s the low down.

The good stuff

The students I saw were eager to learn. When you ask them questions, they engage with you and share ideas. They are capable of noticing patterns and working through problems. They are nice and carry themselves well.

The teacher has students working on problems in groups using whiteboards and they mostly work well together. They are on task almost all of the time.

These students are also willing to put their heads down and do some work. The teacher has a good command of his classroom, as well as rapport with his students.

The “getting there” stuff

The problems students are working on were end-of-the-chapter exercises, which mostly fail the Dan Meyer litmus of giving too many sub-steps. My sense is that problems being pulled are somewhat also random. For a new teacher, this is a OK starting place, because at least he is using textbooks as a resource for getting students to do things. It’s not the worst he could do, but we’ll want him to start choosing problems not just because they are on topic but because they help students make contact with problems that will nudge them forward in their learning.

The students are basically just playing plug-n-chug games with equations–part of this is because the problems are built this way. But another reason is because we have a new teacher who doesn’t know yet how to emphasize reasoning and understanding of concepts. When I asked students what the symbols mean, they didn’t know. The good things is that students will engage with you and work out the meaning in their group if you ask them about it. Someone needs to be there to ask good questions, and that’s something we’ll want to help this teacher with.

Students are working in groups, but those groups work in complete isolation of the broader classroom. There are many missed opportunities for students to learn from each other across groups. I saw lots of examples of students solving the same problem different ways, or coming to insights that either could or really need to rise to the level of the classroom. The way it played out was early in the day was that when students finished, they got assigned a new problem. Once again, not the worst things students could be doing. They are getting a lot of chance to practice problems and figure it out with peers, but we’ll want to eventually leverage what happens during problem solving for deeper learning.

The teacher at one time unintentionally steered a group of students away from a productive and correct solution path, because I think he couldn’t quickly makes sense of what students were doing and anticipate where that would lead them. This is one the hardest things to develop for teachers and is where strong content knowledge and pedagogical content knowledge is important. The ability to quickly listen, understand, assess, project into the future, and decide is one that comes with practice, practice, practice.

My overall advice would be something like:

Things are going pretty well. He is engaging students in problem-solving and having them work in groups. They are engaged and willing to put in work. The classroom also just has a nice feel to it–students are comfortable and so is the teacher. There are a lot of positive things to build on.

While students are engaged in doing problems, there needs to be more focus on concepts and understanding, not just equations. Some ways of beginning this is getting them to use multiple representations-graphs, tables, diagrams. This will help build connections. But asking them to explain their thinking–what they are doing and WHY–is always important.

With small groups, be patient and listen to your students as they work, and then ask questions about what they are doing and thinking. Figure out what they know and what they don’t know before you offer some help. Also, it helps to have some ideas ready about how to extend these problems for those groups that finish early.

Have students share some of the work they are doing so they can learn from each other, and so you can help them make connections. You don’t have to do this all the time, and you don’t have to have every group share when you choose to do it. This helps build community and also makes everyone responsible for teaching and learning.

Overall, the fact that this teacher is looking for help, for ideas, and mentoring is a promising thing. There is a lot of good things to build on, and I’m excited to see where he goes.

I’m thinking today about average velocity again, because on a certain day next week, I have to talk about average velocity. I have decided to talk a little about why average velocity isn’t typically found by adding up velocities and dividing that sum by the number of velocities you add up.

But on the very next day, I’m going to have to use this formula

Vavg = (vf + vi) /2

And I’m going to have to explain why in this case, it looks like we are just taking some velocities and dividing by the number of velocities we have–the very thing I said they shouldn’t do the day before.

So, it’s time to think about averages, again. Let’s say I have these numbers 1,2,3,4,5,6.

I can find the average of these numbers by taking (1+2+3+4+5+6) /6

But that is equal to 21/6 which is equal to 7/2 which is equal to (6+1)/2, which is just the first number added to the last number divide by two. This can, of course, be generalized to any sum of sequential numbers, and even equally spaced numbers.

This is a lot like Vavg = (vf +vi) /2, where you take the initial velocity add it to the final velocity and divide by two. This expression for the average velocity is true only for an object moving with a linearly changing velocity, similar to the way that 1,2,3,4,5,6 was a “linear” sequence of numbers.

I’m not sure it’s worth going into all this, but I just think it’s funny when you tell students not to do something, and the very next day you do that thing.

 

Based on today’s discussion about scientists’ notebooks, I drafted the following rubric for assessing students’ notebooks in class, following the guidelines of a lesson over at SGSI.

Personal Relevance: Does your notebook demonstrate that are you engaged and thinking rather than simply going through the motions of copying down notes and data?

Requirement	page #s – tell me where to look	Grade – I’ll do this
Evidence of your ideas and their progression: this should include things that you do understand, things you don’t understand, and show the process of your thinking.
Attempts to interpret, explain, or draw inferences from observations–not just observations and data.
Questions that arise as you conduct research and your attempts to answer them.
Evidence of personal expressions, personality, individual style, engagement or creativity
Sketches, drawings, or diagrams that convey thinking, capture observations, or depict what was done.

Public Relevance: Is your notebook useful and understandable for others familiar with your research. Can they follow the sequence of ideas, experiments, and data?

Requirement	page #s – tell me where to look	Grade – I’ll do this one
All days are dated and pages are numbered
There is some organization (such as labels or titles) among the “chaos”.
There are references to peers’ work or ideas when they influenced your own thinking.
There are details and accurate accounts of what you observed, such that your findings are well documented
There are detailed, accurate descriptions of procedures, making it possible to go back and repeat or “troubleshoot”

Today we discussed our rights and responsibilities as students and instructors.  In 2-3 weeks, we’ll revisit our list to both assess how we are doing and decide if any changes needs to be made.

Below is the full list of what they came up with, but my two favorite are rights they gave me.

“To change the curriculum and steer the class in productive directions”

“To reject unacceptable work”

Students Rights:

To get help–either from peers or the instructor
To share ideas and opinions and to feel comfortable doing so
To know and understand what’s expected of them
To have constructive criticism
To be treated fairly­–both in how we treat each other in class and with grading
To know of changes to class or class policies in a timely manner
To ask questions
To be yourself

Student Responsibilities:

To be respectful of others and their ideas
To be prepared and engaged­–to be in attendance, to have work completed, and to listen
To be come to class with an open-mind and willingness to learn
To ask questions when you don’t understand or are unsure
To participate and function well in groups
To come in with a good attitude
To contribute to a safe learning environment–both emotionally and physically

Teacher Rights:

To change the curriculum and steer the class in productive directions
To have the class’ attention and participation
To be respected
To expect that students will do good work
To reject unacceptable work

Teacher Responsibility:

To be available
To make class fun and engaging
To maintain a safe learning environment–both emotionally and physically
To help prepare students to be successful in ways they are expected to be
To teach ideas that are useful and/or meaningful
To encourage students
To give feedback in a timely manner
To treat students as adults and as individuals
To be on time and to not let class go over the scheduled time

In this post, I just want to motivate why a gradient gets complicated in a non-euclidean space, and in a following post I’ll work through how to take a gradient. The reason we care is because to find the force from potential energy, we need to take a gradient.

In a euclidean space with cartesian coordinates, the gradient can be taken this way

Grad(U) = (∂U/∂x) x-hat + (∂U/∂y) y-hat + (∂U/∂z) z-hat 

One way of interpreting the operation ∂/∂x is that  it “nudges” whatever the scalar function along in the direction of x, and then looks at how that things change. It also keeps track of how far the nudging has occured, and it takes a ratio.

This “nudging” is really important to get a handle on. See, in a euclidean geometry described by cartesian coordinates, a nudge in the x-direction is always the same nudge no matter what; and a nudge in the x-direction never ever causes a nudge in the y-direction or the z-direction. That’s actually what this metric

ds² = dx² + dy² + dz²

tells you. The 1 in front of dx²  tells you that the dot product dx.dx is a constant of unity. The fact that there is a (hidden) zero in front of dx.dy and dx.dz tells you that nudges in x are not influenced by or influence nudges in y or z directions.

But if you remember back to my first post, we had this metric:

ds² = (1+α²) dx² + (1+β²) dy² + αβ dx dy

In this metric, we have to three things to consider when nudging:

(1) A nudge in the x-direction is not always the same nudge depending on where you are, unless α is a constant; and similarly for the y-direction.

(2) A nudge in the x-directions isn’t the same nudge as a nudge in the y-direction, unless α = β.

(3) A nudge in the x direction will co-occur with a nudge in the y-direction as along, unless both α and β are not equal to zero

Another way of thinking about this is like this, imagine you had the vector x-hat, which is a unit vector. Imagine you tried to drag x-hat in the x-direction. In a euclidean space, x-hat would still have a magnitude of 1, and x-hat would still be facing in the x-direction. However, in a non-euclidean geometry, two things can happen. One kind of that can happen is that x-hat can grow or shrink. That growing or shrinking is caused by terms like α² or β². Another kind of thing that can happen is that you’re x-hat vector, which you dragged in the x-direction, is all of sudden pointing a little bit in another direction. That’s what the term αβ does. It twists your unit vectors around.

This growing, shrinking, and twisting is what we’re going to have to keep track of in order to take a gradient. How we keep track of this is the subject for another post, but I hope you get the big picture–a nudge is not just a nudge but a nudge with a growth and a twist.

OK, so building off my last post, the question I want to work through is, “How do free particles move in spaces that are non-euclidean?”

Let’s use Andy’s suggestion of an object sliding around on a parabola, and we’ll project that parabola onto the x-axis, to see what motion is like on the 1D projected parabolic space. But let’s take away gravity, because we want to consider free particles that are subject to no ‘external’ forces.

We know of course that in the 2D world, the particle would move with constant speed from one side of the parabola to the other. The reason it would move with constant speed is because, without gravity, there is no way for the object to change its kinetic energy, or direction for that matter.

What I want you to imagine is you are looking down into the parabola from above, and you can only see what’s happening in the x-direction. It’s like you have no depth perception. To you, the ball would seem to speed up as the ball goes down the parabola and slow down as it goes back up the parabola. That’s because some of the speed is “hidden” in the y-direction, which you can’t see. Keep this in mind, because our mathematics has to jive with this intuition.

Getting the Projected Parabolic Metric

So let’s take the parabola y = 1/2 x²

As I said in my previous post we typically measure distances like this

ds² = dx² + dy²

But we can transform this 2D euclidean space to our 1D non-euclidean space by using the fact that

dy = x dx

Plugging that into ds² = dx² + dy² gives

ds² = (1+ x²) dx²

Recognize that the term “x” is just the value of the slope of parabola at the point x, so it’s completely consistent with what I’ve talked about in the first post.

Newton’s 1st Law in the Projected Parabola

Now, I’ve already established that the object moves with constant speed in the 2D euclidean world. So let’s set a value for that velocity and just call it “v”.

With a velocity v, the distance in the 2D geometry would be s = v t or ds = v dt, where s is measure along the arc length of the parabola.

This allows us to write

v² dt² = (1+ x²) dx²

which I’ll rearrange to be

(dx/dt)² = v² / (1+ x²)

or

dx/dt = v / Sqrt (1+x²)

Here is where we should check out intuition, because the value dx/dt is the velocity in the x-direction. I said that the ball appear faster near x =0, because there none of the speed is hidden in the y-direction. Plugging in x =0, we get dx/dt = v. That’s good, because that’s when none of the velocity is hidden. If we take the limit as x gets really large, the velocity goes to zero. And that also makes sense because the parabola gets so steep that you can’t see any of the velocity–it’s all hidden in the y-direction.

OK,  what we might want to do is write Newton’s 1st laws for a 1D parabolic geometry. So what we want is to rewrite this expression so that it looks like

d²x/dt² + Correction[x, dx/dt] = 0

And we can do this by taking a time derivative  of our equation for dx/dt and rearranging.

d²x/dt² – v * d/dt [ 1/ Sqrt (1+x(t)²)] = 0

d²x/dt² + v * x / (1+x²)^3/2 *dx/dt = 0

Now this is pretty ugly, but it puts it in a form like Newton’s 1st law. Our corrections term is the messy part, because it is non-linear function of x and also depends on dx/dt. It might be tempting to think of this whole term as  velocity and position dependent “force”, but we don’t have to. I want to stress that this is just how particles move in our 1D projected parabolic space without forces. I’ll show you later how we can add forces.

The truth is, it would have been prettier to not take that derivative, and cast this in term of energy. One way of doing this is to multiply our expression for (dx/dt)² by 1/2 m, to get

1/2 m (dx/dt)² = KE / (1+ x²)

where KE is just a constant.

Adding Back in Gravity (the easy way)

To add back in gravity, we just have to realize that KE is no longer a constant in the 2D Euclidean space. But since the total energy in 2D euclidean space is still constant, we can write

KE = Total Energy – PE = TE – mgy = TE – mg (1/2 x²)

1/2 m (dx/dt)² = (TE – mgy) / (1+ x²)

1/2 m (dx/dt)² = (TE – 1/2 mg x²) / (1+ x²)

where TE is just a constant expressing the total energy in the 2D Euclidean space.

I’m pretty sure this works. I suppose we’ll see.

Adding Back in Gravity (the hard way)

This will be the subject of another post, but basically we’ll start with our Modified Newton’s 1st Law expression, and we’ll have to figure out how to take a gradient in a non-euclidean space.

As an aside,  I’d like to point out that SO far, we haven’t had to use calculus of variations or Langrangians.

I’m a big fan of thinking about wrong ideas. One wrong idea I’ve been spending a lot of time thinking about for over a year is one that I came across while working through a piece of curriculum intended for physics majors to learn more about this relationship

F = –Grad(U)

The materials had students think about a rock on a mountain, and how the force on the rock would be related to the slope of the mountain at that point. The  materials went on to have students think about a topographic map, where the height of the mountain could be described everywhere with some function z = f(x,y) and the potential energy by U = mgz. They had students eventually build up to considering this expression

– mg * [(∂f/∂x) x-hat + (∂f/∂y) y-hat ] ,

which would appear to be the a “projected gradient” of some sort.

In this post, I want to explain why this expression is (1) not a gradient and (2) not a force.

So, let’s imagine an object moving on some randomly shaped “frictionless” surface.

And let’s say that we can describe this surface as before, with z = f(x,y). This surface could be a ramp, or a bowl, or a complicated ice sculpture. It doesn’t matter, but I want to consider the case that the object can’t fall off or fly off. Let’s also imagine that gravity is involved and that the gravitational potential energy of the object is

U= mgz, as was assumed in the rock on the hill case.

So by substitution we can write

U = mg f(x,y), as was done in the curriculum.

Now if there was no surface involved (i.e., the object was just free-falling in gravity), then we could certainly relate  the force due to gravity  to the potential energy in the following way

 Fg = -∂U/∂z z-hat= -mg z-hat.

However, that’s not going to be true with our surface constraining the motion. Given this, however, I do agree it’s tempting to consider the expression

– mg * [(∂f/∂x) x-hat + (∂f/∂y) y-hat ],

which involves taking some the partial derivatives of U with respect to x and y (instead of z). The question is, is this a force or perhaps a projection of a force? It’s certainly tempting to think that this is, because it looks like the gradient of the potential, and it looks to be happening in the x-y plane.

It turns out this isn’t a force, because taking the partial derivatives in this way does not constitute a gradient. The reason it’s not a proper gradient is interesting (to me) and has to due with the fact that taking partial derivatives like this only makes a gradient in the special case of a euclidean geometry. To do so, I will have to convince you the transformation of U(z) to U(x,y) is very different than what it at first seems.  So, we’ll have to back track a bit.

See, in a 3-dimensional euclidean space, we measure distance by the following metric:

ds² = dx² + dy² + dz²

But what we have is different. We have a 2D surface embedded in a 3d space. Given that  z = f(x,y), we can write the differential dz = (∂f/∂x)  dx + (∂f/∂y) dy and substitute that into our 3D metric to get the following metric for the 2D surface:

ds² = dx² + dy² + [(∂f/∂x)  dx + (∂f/∂y) dy]²

This can be simplified and rewritten as

ds² = (1+α²) dx² + (1+β²) dy² + αβ dx dy

where I’ve just named ∂f/∂x  as α and ∂f/∂y as β.

OK, so now have this 2D non-euclidean surface described by the above way of measuring distances. It is weird, because you have these correction factors such as α² and β² that augment the distance traveled depending on the slope of the surface where you are measuring. We also have this weird cross term that involves dx dy. These terms are what make space non-euclidean. It’s like your ruler grows and shrinks and even twists as you move around the surface. Or, if you will, those corrections exist because we are only looking at x and y and the correction factors add back in the distances in the z-direction.

So the following questions arise:

1. How do free particles move on such surfaces?
2. Can we calculate forces by taking the gradient of the function U = f(x,y)

OK. OK. Let’s step back again and clarify why these are the two important questions:

In a Euclidean geometry, free particles can be describe by the follow equation

m d²r/ dt² = 0 → This is just Newton’s first law.

and many types of forces can be calculated  using by F = – Grad (U)

This allows us to write Newton’s 2nd law as,  m d²r/ dt² =  –Grad (U)

All of this works for Euclidean geometries and for forces described by conservative vector fields–but not all conservative forces, just forces described by conservative vector fields. There’s a subtle difference, because Normal forces are conservative (i.e. they do no work), but they cannot be described by a conservative vector field.

OK. So, what we’d like to do is write a similar equation for particles on our surface. The problems we’ll end up running into are these:

1. Free particles on our 2D surface can’t be described by d²r/ dt² = 0, and
2. We’re going to have to be smart about how we take our gradient.

I want to take up these two obstacles in two other posts. So, next I’ll talk about free particle motion in non-euclidean spaces. And then I’ll talk about gradients and external forces in non-euclidean spaces. Hopefully, I’ll get to a fourth post where I talk about why classical mechanics and general relativity are the same thing. And last, I’d like to talk about the mistakes students made in this curriculum, especially the ones that got me thinking about these issues so deeply. Wish me luck.

I’m a big fan of either using 10 m/s/s or 22 mph per second for g, and giving students reasonable numbers when they are doing exercises that involve projectile motion. The real world won’t be so kind when they take data, so I want to start them off where they have a fighting change of reasoning proportionally on the basis of concepts. I’m trying to fight the “iterative plug-n-chug” approach, by doing things we can think through.

But, in one of my classes, I’ll need to be using 9.8 m/s/s, because that’s what they’ll have on their exam. And I won’t have reasonable numbers always, because the problems I’m supposed to provide example are not picked by me. So, I have decided to do a lot of  “figure out a number you know is too high”, “figure out a number that is too low”, and “make a good guess”, before doing any plugging into equations.

Take for example, this kind of question I am supposed to work through as an example:

A ball is thrown vertically upward with a speed of  7.2 m/s. On the way down the ball hits a flag pole that’s 1.2 meter above from where it was thrown.

It’s a goalless problem, so I’ll have to decide something to solve for. Let’s go with, “How long will the ball take to get to its highest point?” … Here we go. Well, we know that the ball loses 9.8 m/s every second, but it only needs to lose 7.2 m/s. That must mean that the time to reach its highest point is less than 1 second. How much less? Well, it’s certainly greater than half of a second, because half a second would be  time to lose 4.9 m/s.  So, a number that is too high is 1 second, and a number that is too low is 0.5 s. I would guess it’s probably pretty close to 3/4 of second, because 7.2 seems pretty close to in between 4.9 and 9.8.

What’s the actual answer? About .73 seconds.

OK. Now on to another question. “How high will it go?” Well, the average velocity of the ball during the trip will be right in the middle of its starting velocity (7.2 m/s) and its ending velocity (0 m/s). That makes its average velocity 3.6 m/s–I can divide by two in my head. Given that I’ve concluded in the first part a travel time of 3/4 of a second, this must mean it must have traveled less than 3.6 m, because it would have taken an entire second to travel 3.6 m, and it didn’t have that much time. My reasoning is that it should go 3/4 of the way to 3.6m in 3/4 of a second. This is actually pretty easy to reason about, because 36 = 9 * 4. So, I’d say it would have gone about  2.7 meters during that time.

OK, so what’s the actual answer? 2.6 meters

OK, so how much time to hit the flag? Well, it took 3/4 of a second to go up. It would have taken another 3/4 of a second to go down, but it didn’t quite go all the way down because it hit the flag. So, How far did it get? Well, it got a little over half way down (2.7-1.2 = 1.5). Let’s just call it half for good measure. Now it might be tempting to think it took half the time to go half way down as the whole way down. That would be true if it was moving at constant velocity. But it’s moving slower earlier and faster later, because it takes time to speed up. Given that, we know that it must take MORE than half the time to cover half the distance, because that’s the slow part of the motion. By that alone we can say that it must have taken more than 3/4 + 3/8 = 9/8 of a second. So we have a lower bound of 1.1 s and an upper bound of 1.5 seconds. Let’s just guess that’s it took 0.5 second, because that’s more than half of .75 seconds.

That would mean the whole trip should be about 1.25 s = 0.75s on the way + 0.5s on the way down seconds.

The real answer? 1.2 seconds.

OK. So, some of this is a bit much. But, I didn’t have to use equations.