The Five Best Lightsaber Fights in Star Wars

The great thing about having a personal blog is that I can write about whatever makes me happy. Today, it’s Star Wars.

Actually, this was a question that my older son asked. What are the best lightsaber battles? I love these questions—they don’t really have just one answer so you can argue over who is right with no real winner.

Let’s get to it. If you don’t agree with this list—I’m ready for your complaints.

5: Darth Maul & Savage Opress vs Darth Sidious (The Clone Wars)

Just to be clear, this is from the Clone Wars animated series. This is my son’s favorite (he’s a huge Clone Wars fan). It’s pretty cool because it shows how powerful Sidious can be when he wants. Also, bonus points for the Dark Saber.

Double bonus, here is my son’s reenactment of this battle. Brace yourself.

4: Luke vs. Darth Vader (Episode V The Empire Strikes Back)

Yes, it’s true that this fight isn’t as dynamic as those in the prequels or the Clone Wars. I get that. Also, it’s possible that I just like this one because I’m older and this battle was so epic when it came out. But you have to admit that the cinematic quality here is awesome. Love it.

3: Qui Gon and Obi Wan vs. Darth Maul (Episode 1: The Phantom Menace)

This could be in the top five just because of the music. It’s also the first time we see how crazy a real lightsaber can get. I mean, before this all we saw was old men, half robot, and young mostly untrained warriors.

Also, it’s got a nice two vs. one AND a double light saber.

2: Obi Wan vs. Darth Vader (Episode IV A New Hope)

OK, this might not count. This is a fan-made redo of the battle between Darth Vader and Obi Wan on the Death Star. I guess it’s not canon in the Star Wars universe—but I can’t help myself. If this was how it originally happened, I would put this at number 1.

Oh, I was watching Episode IV the other day. When it came to the Vader vs. Obi Wan fight, I paused the movie and switched to youtube to watch this version.

1: Obi Wan vs. Darth Maul (Rebels)

You might not like Star Wars Rebels—I get that. It’s sort of like a combination of Aladdin and Star Wars. However, there are some really great parts in this show. When Darth Maul finds Obi Wan and tries to kill him—this is just the best. It’s not about the light saber action, but it’s still just the best.

Blogging Update

Hello there.

Yes, I haven’t posted here in a while. But to make it up to you, I’m going to show you a picture from my drone.

This is a canal in New Orleans. My older son had a soccer game right next to this, so during warm-ups I did a little bit of photography. Actually, these are really weird. You are on ground level and you see a big long hill (there really aren’t many hills in New Orleans). When you walk up the “hill” you see water that could be at a higher level than the ground.

Yes, it’s not a hill. It’s a levee.

It’s definitely odd.

Now for some other random updates.

  • I have been busy as usual. I picked up a couple of freelance posts for a UK magazine. In the edits, all my “meters” were changed to “metres”. I thought that was fun.
  • Blogging on WIRED went pretty well this week. I had two posts – one on a rough estimation of the amount of carbon dioxide capture from planting trees. The other was an estimation of Death Star pieces on the surface of a planet.
  • Another thing I have been working on: making better youtube videos. I have made some modest gains, but I still need more work.
  • I was thinking about doing a youtube live stream event, but I’m afraid no one would show up.
  • Finally, my oldest daughter moved to Japan. I’m glad she made it there.

Notes and Comments on AAPT Summer Meeting 2019

Here are some things I need to share regarding this meeting. Overall, it was great to see everyone. As usual, the conversations were the best. I regret that there were some people I did not get to meet or talk to—maybe next time we will meet up.

Blog vs. WIRED

One question that came up multiple times was about this blog vs. my posts at WIRED. How do I decide where a post goes? OK, here is my explanation.

In the beginning, there was a blog. A blog was super informal and free form and alive with comments. It was like the 60s and I was a hippie. A physics hippie. I don’t know if this early blogging era was like the 60’s or maybe the wild-wild west. Well, the comments eventually turned into the wild west with a shoot out at the O.K. Corral.

When I moved to WIRED, everything was the same except it was at a different site. But I’ve been there for a LONG time (9 years?) and things evolve. My posts at WIRED are more edited and geared for a specific audience. That’s not bad, it’s just different. I don’t think I can just write whatever I want like I did in the old days. No more random posts that just talk about my cat (I don’t even have a cat).

So, that’s where this blog comes in. It’s a place where I can post whatever I want and no one can stop me. These are the kinds of things you will find here.

  • Random posts (like this one) that are just an outlet for me to write stuff and tell stories.
  • Explicit educational material. If a post needs too many equations, I would rather put it here. Many WIRED readers (while very education) don’t really get into all the equations. Also, since WIRED is paywalled it makes it more difficult for educators to access the stuff (in the off chance that they might find it useful).
  • MacGyver science notes. Oh sure, I post some MacGyver stuff on WIRED—but I really don’t think they want to see 50 posts on different episodes. So, those are here.
  • I think that pretty much covers it, so I don’t even need this last point.

In the end, I apologize for the confusion with the two blogs. Oh, actually there are three. I recently wrote a post on OneZero Medium (analysis of a car crash from Stranger Things) also. Not sure how much I will write there—but it’s still me.

What’s up with all the drone videos?

Yes, I have a drone. I love my drone. I can only hope my drone loves me as much as I love it. Honestly, I am honored that you even noticed my drone videos.

Oh, wait. You haven’t seen them? I can fix that.

In case you are curious. This is a DJI Spark. Great drone.

More Comments

Here are some more short comments.

  • Meeting with Bruce Sherwood and Ruth Chabay was great. I wish I had a picture with both of them (I did get one with Bruce though).
  • Bruce made this epic comment in regards to numerical vs. analytical calculations. People claim that analytical solutions are better because you can solve a problem in terms of known functions like sine and cosine. But how do you find the value of the cosine function? YUP – numerically or in a table or in an infinite series. So, in a way all solutions are numerical. Win for numerical.
  • The other deep thought by Bruce was a discussion on his AJP on energy. Read that paper. This sums it up. You can not find the work done by friction. Friction is crazy hard. I think I might write a WIRED post on this.
  • Eric Ayars had an excellent presentation on chaotic systems. One system was a bouncing ball on a moving floor. I wonder if there is a case where the ball just stops—this could happen if the relative collision speed of the ball and floor is zero.
  • The 30 demos in 60 minutes was pretty good. I love these things. Even though I’ve seen many of these demos before, I always find something new. Here is their site http://30demosin60minutes.com/
  • I went hiking. It was super hot, but I had a great time.

It’s Just One Semester at a Time

This is really for students—but maybe it applies to you also.  If so, I’m happy about that.

So there you are.  There’s still a month left in the semester (or quarter) and you are just plain burned out.  You have not motivation to study and your last test score wasn’t quite what you expected.  That thought creeps into your head—maybe you just don’t belong here.

NO. Don’t listen to that voice!

Yes, we all have that voice.  It’s in us all.  It’s the voice of doubt.  You can get through this—surely you can.

Let’s stop for a moment and consider something else.  Suppose you are in a race.  It’s a long race—maybe it’s a 10k.  You haven’t run this far before and you are worried about finishing last so you start off with a quick pace.

Oh, now it’s up to the 8 kilometer mark and you have lost it.  You have to stop.  You can’t keep up this pace anymore.

Has this happened to you in a race?  It has to me (and I hate races).  Of course the problem in this situation is the pace.  You can’t start off too fast or you will run out of energy.  You have to start off with a reasonable pace that you can keep up with the whole time.  It is indeed odd that starting off slower gives you a faster overall speed—but it’s true.

Back to studying.  You can see where this is going.  If you start off at a whirlwind pace at the beginning of the semester, you are going to run out of steam.

Here are some tips for taking care of business during the semester.

  • You don’t have to be perfect in all (or any) of your classes.  That’s like assuming you are going to win in every race.  No one wins all the time—and this isn’t even a race.  It’s not a competition.
  • Take some breaks.  I’m not saying you should just sit around and chill, but if you work all the time your brain can’t process stuff.  Do something fun.  Go see something.  Hang out with friends.  These are the parts of college life that will have a huge impact.
  • Work with others in a study group.  This means you will help others and this means others will help you.  Both of these things are super useful.
  • Exercise.  Go for a walk or hit the gym.  Personally, I like to run—and I don’t use earphones.  Just use that exercise time to sort of meditate and let your brain unwind.
  • Need help?  Get help.  There are plenty of people to help you.  Go talk to your professor (they are most likely nice). Talk to your friends and family.  If you feel like things are getting out of hand, there are probably support services at your university.

Finally, maybe you like dogs.  Go find a dog and pet a dog.

Physics and Education Majors

There is this course.  It’s called Physics for Elementary Education Majors (PHYS 142) – maybe that’s not surprising.  Anyway, I really like this course – it’s awesome.  Let me tell you a little about the history and future of this course.

According my email archive, I think this course was created in 2003.  Ok, technically it was created before that but 2003 is when we started offering the course again.  Actually, the fact that the course already existed made it much easier to get it going.  If you have ever been part of a university curriculum committee, you know what I mean.

We created the course for the College of Education.  They needed a science course for their elementary education majors that satisfied some particular component of NCATE (the accrediting agency for Colleges of Education).  I honestly don’t know (or can’t remember) what specific thing the course was supposed to do – but there it was.  This course was perfect for them.

The first semester I taught this course, I used the Physics by Inquiry (McDermott) curriculum.  This curriculum was especially designed for education majors – and it’s quite awesome.  However, there was one problem – maths.  There isn’t a ton of math in PBI but there is enough to make students panic.  I think they should indeed work through their issues with math, but it was causing problems with the course. Note: I tell students that they shouldn’t say “I’m not a math person”.

After math troubles, I decided to switch to a new curriculum.  At the time it was called Physics for Elementary Teachers (PET) but was later changed to Physics and Everyday Thinking (also PET – by Goldberg, Otero, Robinson).  Here are some of the awesome features of PET.

  • Student learning based on evidence collected (not authority learning from the textbook or instructor).
  • Explicitly includes ideas about the nature of learning.
  • Emphasis on model building and the nature of science.
  • Includes children’s ideas about physics.
  • Math isn’t a barrier.
  • OH, the best part.  The new version of the curriculum is called Next Gen PET.  This version explicitly aligns with the Next Generation Science Standards.  This should be a huge win for the College of Education.

Honestly, it’s great stuff.  Oh, there are still problems.  Students get caught up in the whole “why don’t you just tell us the answer?” thing – but I can work around that.

But like I said – this is the course that we have been teaching for 15 years (wow – even writing that is incredible).  This course was designed for the College of Education.  We typically have been teaching three sections of the course each semester with an average of about 25 students per section.

PHYS 142 Today

I accidentally discovered something recently.  The education majors informed me that PHYS 142 is no longer required in the curriculum.  What? How can that be?  Yup, it’s true.  The new science requirements for elementary education majors have the following three courses:

  • Biology 1
  • Biology 2
  • Earth Science

That’s it.  I’m sure those are fine classes – but they miss a big thing.  They don’t emphasis the nature of science.  In fact, I suspect that these three classes might actually decrease the students’ understanding the nature of science.  Since these three courses have quite a bit of memorization elements in them, students might come away with the belief that science is about facts and not model building.

Yes, I’m not too happy about this.  Not only do I think this course is perfect for education majors (who will be the first to introduce science to children in many cases).  I also genuinely enjoy teaching this class. It’s great to interact with students and see them increase their understanding.  There’s nothing quite like being there when a student starts putting different ideas together.  It’s great.

On a logistical note, this course as some other huge impacts.  First – teaching load.  If we have 3 sections of this course, that would be 15 hours (it’s a 5 contact hour course).  Getting rid of the course will lose 15 contact hours for the department.  That’s one instructor position.  That sucks.

Oh, also I usually teach this course during the summer session. That’s going to suck to not have this.

 

 

Fixing Stuff Isn’t So Bad

Today it started with a lose bolt that holds down the toilet.  Actually, the bolt was gone – beyond gone.  Rusted away.  I figured the flange thing that the bolts go into was probably busted too, but you can’t tell until you lift up the whole toilet.

When starting a project like this, I always go to youtube first.  What a lucky time to live in that we have a resource like this.  Of course it wasn’t just the broken bolt.  I needed to replace the wax seal too.  As long as I’m working on the toilet, I should also replace that tank fill valve that was slightly leaking.

Oh wait! When I turn off the water to the toilet with the shutoff valve, that is leaking too!  After a trip to the hardware store and multiple rounds of turning off the water to the whole house, I finished the job.  The seat doesn’t rock, the shutoff valve doesn’t leak, and the fill valve stops when fill.  Oh, how about a quick advertisement for those quarter-turn shutoff valves?  My house has these “multi-turn” valves that are like the kind on your outdoor water hose.  I’ve already had two of these leak – so that’s not so good.  Also, when the toilet overflows (it happens), the quarter turn is faster to shut off.

 

IMG_5448

Photo: Rhett Allain

I’m just going to say one other thing about toilets (this is not a toilet post).  It seems like a pretty simple job – but once you get confined in space, everything gets just a little bit more complicated.

OK, but in the end – it’s finished.  In the end, I’m pretty happy.  I feel like I accomplished something.  I fixed something and made it better.  I feel human.

I wasn’t always like this.  I remember in my early adult years I would think:

Why can’t they just make toilets that don’t break?  For that matter, why not make a lawn that doesn’t need mowing.  Oh, how about clothes that don’t need washing?  All of this cleaning an fixing stuff is just taking away time from more important stuff.

That was the old me.  The new me doesn’t mind these chose so much.  Oh, I still get bothered sometimes.  When the lawn needs mowing and it’s super hot (or won’t stop raining) or when something breaks and you JUST KEEP MESSING UP.  Yes, those times are frustrating.

But I’ve come to accept that we live in a world that increases in entropy.  If you leave stuff alone, it will just mess up and eventually break.  What makes humans so awesome is that we can fight this tide of increasing disorder.  We fix things.  We clean things.  We are humans.

 

 

A Quick Note: Packing for College

My oldest daughter is currently in the process of getting her stuff together.  She is moving away for her first year of college.  Hope everything goes well, but I keep thinking of this scene from The Hobbit.

Yes, in college you will have to do without a great many things.  Honestly, that is part of what makes the whole college experience so great.  It’s not just about classes, but all the things in between (if it was just about classes – it would make more sense to stay home and take classes online).

No, you won’t have everything perfect in college (or in life really).  You won’t have the best shoes for a particular event.  You might have to wear the same pants more than once.  You are going to have to share a room with someone – and a shower too.

But you know what?  It’s not just college – real life is like that also.  It’s not about making everything perfect, it’s about living with what you have.  You can never have a perfect life – unless you learn to enjoy the imperfections that life throws at you.

 

A Cave Diving Story

I want to write about some things that happened in the past – mostly so I don’t forget.  In this case, I’m going to describe a particular cave dive that sticks in my mind.  Now, there is a warning.  This might not be exactly what happened – but it might also be true.  It’s probably the way it happened.  Let’s begin.

In case you aren’t familiar with cave diving, the main idea is to explore a cave that is completely underwater.  You start outside the cave and you are already underwater. This is not spelunking – this is cave diving.  We mostly went into springs in the panhandle of Florida – that was our roaming territory.

Just for fun, here is a picture of me with my cave diving gear.

Please don’t complain about the setup.  This was a picture from my cave diving class.  I learned quite a bit after this and changed things up – but still you can get an idea of what it was like.  That was my friend’s van that we used to drive to different caves – it was pretty sweet.

Ok – now for this one particular dive.  I am pretty sure this was at Twin Cave.  Twin Cave is in the middle of Merritt’s Mill Pond in Marianna Florida.  The only feasible way to get here is with a boat (unlike most cave diving in which you just walk into the water). It’s called “Twin Cave” because it has two entrances right next to each other (underwater).  If you have a boat, you can use a tiny little wood platform connected to two trees to set up stuff.

My dive partner (Trey) and I hitched a ride with Pat Watson.  He had a little john-boat and ferried us to the cave.  He then went back and picked up two more divers – Pat was teaching them how to cave dive and this was one of their “check out dives”.  Oh, I should add that at one point there was so much stuff in that little boat that it was barely above water.  One tiny wave and I think the thing would sink.

Now for the dive plan.  Go into Twin Caves and get into the lower levels.  In this case, we would use diver propulsion vehicles (DPV) to get through the easy part and then proceed in the lower part by swimming.  The DPVs are awesome (also called scooters).  That’s the truth.  I always wanted a scooter – but I never got one.  Fortunately, my dive buddy had just recently upgraded his DPV for a bigger one.  That meant that he still had his old one.  BOOM – scooter time,

You’ve seen these scooters.  They are basically small torpedo looking things (not exactly) with an electric motor.  You connect a line from the scooter to you so that you can “drive” with one hand.  That’s not really important, but I just wanted to say that.

Ok, so we start off on our dive.  I seem to recall that it’s sort of tricky getting into Twin Cave.  The main entrance is a vertical shaft – but then you have to turn horizontal right when you get in.  If you don’t turn correctly, you hit the the bottom of the cave.  Most cave bottoms have this silt stuff – if you hit it you change the cave water from pristine clear water to brown soup that you can’t see through.  So – don’t hit the bottom (this will be important later.  Oh, it’s even harder to get in this cave when you are trying to do with with a scooter.  We probably had a stage bottle too.

I guess I should say something about stage bottles – for completeness.  A stage bottle is an extra scuba tank that you mount on your side.  You use this stage bottle for the first part of the dive – until it gets 2/3 full.  At that point, you take the tank off and leave it on the line (there is a line running in the cave).  You then proceed on the dive using the two tanks on your back.  Using stage bottles lets you go much farther than just with double tanks.

Now back to the dive.  The first part of the cave is pretty cool.  It’s this big cylindrical cave that we called the subway.  Ok, that might not be true.  In my mind, we called it the subway.  It was super easy to get through with a scooter – just imagine a long straight circular tunnel with a diameter of about 3 to 4 meters (let’s say 10 feet for imperials).  Like I said, this is perfect for scooters.

At some point, we moved down to some small stuff and dropped off the scooters.  There are two important points here.  First, this part was deeper (can’t remember the exact depth) such that we will have more decompression time.  Second, it was low and slow down here – longer time at depth also means more deco.  I’m not going to really talk about deco time right now – but let’s just say that you don’t want to rack up a ton of deco time – it sort of sucks.

So we did our exploration of the lower part.  Now it’s time to get the hell back to the entrance and do our deco.  No more sight seeing – just hurry up.  We pick up our stage bottles and our scooters and scoot out.  Oh, quick note.  There are two things that are awesome to see in a cave.  The first is the warm glow of the Sun when you are near the cave entrance – this tells you that you are almost out.  The second is your stage bottle.  Picking up a stage just says “hey, you just got a bunch more air and a completely independent air supply – you have less chance of dying now”.  It’s awesome.

Now for the whole point of this story.  We are there on our scooters moving out of the cave while back in the subway.  Ahead we see some lights.  It was Pat and his two students.  As we start to get near, something happens.  The inflator on Pat’s buoyancy compensator (BC) just falls off for no apparent reason.  In case you have never been diving, the BC is basically a bag of air that you can fill up or empty to adjust how you float in the water.  It’s super important in cave diving because you want to stay off the bottom of the cave where all that silt is (remember I mentioned this before).

When Pat’s BC failed, so did his ability to stay off the bottom of the cave.  It was weird – he just took a dive straight down.  He sort of looked like a B-17 bomber that had just been shot out of the sky.  I saw him try to make a correction to his buoyancy, but it wasn’t going to happen.  He ended up just putting two fingers straight down to catch his fall (this is what we were taught so as to minimize the silt disturbance).  It didn’t work. His crash produced silt.  A lot of silt.

It was really amazing.  This clear water subway tunnel suddenly turned into a tunnel with a wall.  A wall of silt.  You really can’t see much in this silt – it’s effectively zero visibility (but not as bad as clay in the water).

Since we were traveling along at a good pace with the scooters – it only made sense to slow down quite a bit.  I knew there were three divers inside that wall of silt, but I didn’t know WHERE exactly they were.  So I slowed down.  Gradually clicking the throttle trigger on the scooter to maintain a slow crawl speed.

To my surprise, my dive buddy chose a different strategy – he decided to just keep going at his current speed.  I saw him head straight into that wall of silt – waiting for some type of collision.  But no, he made it though.  It was a bold move and it paid off.

For me, I went through slowly and made it to the other side of the silt wall (which wasn’t very thick).  After that we did our deco and finished the dive.  I sort of recall not taking the boat back to the van and staging area.  Instead, we used the scooters the rest of the way.  It didn’t matter at that point – you don’t have to worry about saving scooter battery or scuba air.

The end.

Picking up trash

There was this very nice lady in our neighborhood.  She would take it upon herself to pick up trash along the main road next to the subdivision.  Super nice lady.

Well, she moved.  She moved to live with her kids (she was older).  So, as a way of respecting her service I went on amazon and ordered a pair of those stick grabber things.  Now it would be my duty to keep that road clean.

The first time I picked up trash, my oldest daughter volunteered to go with me.  There was a BUNCH of stuff.  Crazy stuff.  I think we filled up three trash bags for about a mile of road.

Cigarette butts are all over the place.  Styrofoam cups, beer bottles, vodka bottles, and other various things (some of which I shouldn’t mention).  It was hot and buggy.

But in the end, we finished.  The road looked better.  The next day, I saw a cup. WHAT THE HECK???? I just cleaned up (we just cleaned up) and there already was a stupid cup on the side of the road.  Who does this?

So now, I go about once a week.  I pick up trash.  It’s hot – but I really like it.  It’s like one of those chores that you oddly enjoy.  It’s like vacuuming the floor and it looks all neat afterwards.  Or mowing the lawn, or cleaning the bathroom.  I sort of like these jobs that clearly make a difference.  On top of that, picking up trash is about more than just me – it’s a sort of community service.

Oh, and I get some exercise too.  To summarize: don’t pick up trash on the road – it becomes addicting.

Working notes for my bouncing ball running model

I’ll be honest.  I had some problems getting my bouncing ball running model working.  Oh, here is the model.

https://www.wired.com/story/how-is-a-runner-like-a-bouncing-ball/

Basically, this models the speed of a running human by assuming they are bouncing ball.  When the human impacts the ground, there is some maximum impact force and an impact time.  The impact time decreases with horizontal velocity such that eventually, all the force is used in the vertical direction to keep the human off the ground long enough to switch feet in the air.  The end.

As I was making this model, I took some notes because I couldn’t get it to work.  Here are my notes.  Hopefully you can use this to see how to troubleshoot a program.

Running model notes

I think I mostly have it working:

http://www.glowscript.org/#/user/rhettallain/folder/blog_posts/program/runningbouncemodel/edit

Here is basically how it works.  Two big ideas:

  • Humans can push off the ground with some maximum force.  This force does two things – gets them off the ground and in the air so legs can move and pushes them forward
  • The contact time with the ground is small and gets smaller as horizontal speed increases
  • This means as the human speeds up, the ground force eventually gets to where it can only push up and not forward

Here is what it looks like so far

Here is a graph of speed vs. time

  • This model reaches a max speed of about 3.5 m/s in just a couple of strides – that doesn’t seem right
  • I think my Fv calc is wrong – it gives back the same speed not the needed vertical speed to get the stride time
  • Need to recalcualte Fv based on pfinal
  • If you want to be in the air for ts seconds, then your initial vertical velocity must be -g=dv/dt.  dv=g*dt dv = 2vstart. start=(½)gdt
  • Now to calculate the force. I know tc (contact time) so F = dp/dt = m*(vy2-vy1)/tc – this is the total force = Fv-mg so Fv = that stuff +mg

Something isn’t right.  Here is a plot of position vs. time

It’s getting higher and higher (and going lower – weird)

  • I’m getting stride (in air) times of 0.09 to 0.13 – that’s wrong

 

Ok – I think I know the problem.  I need to set the force push time loop and forget about while human.pos.y<R – I think that’s my problem

 

How about this

  1. Once human hits the ground – calculate Fv, Fx, and tc set tcount = 0
  2. While tcount < tc – set human.pos.y = ground. And set the forces
  3. When tcount = tc, turn off the forces and stop holding the person

 

It appears there is something wrong with my Fx.

  • Fx is some value for the first push – but after that it goes to zero and the Fv is maxed out.
  • Werid
  • There is a problem with both Fv and Fx

 

The problem is the time of impact – it gets too small such that the required force is HUGE

  • How about a min time – and it can’t go lower?

 

Fmax = m*2v/t

t=m*2*v/Fmax

 

I think the problem is that during the contact time, the horizontal force is too much so that the human ends up going faster than the theoretical speed.

 

I can use the time and force and velocity to estimate the average velocity and then recalculate the time

This is the paper

https://www.physiology.org/doi/pdf/10.1152/japplphysiol.00947.2009

It has this plot.

This shows a decrease in contact time with speed

Here is what I get for a fit

This gives a contact time function of

Although this “blows up” at v= 0.  Maybe I should say tc = 0.3612 for v < 2 and this expression for v>=2

End of notes – it finally worked.