# Algebra Review Worksheets for Physics

I noticed in my blog stats today that someone found my site because they searched for “algebra review worksheets for physics.” I didn’t have anything like that here, so I thought I would add some, in case anyone else was wanting to brush up on their math skills before taking physics.

I do have a page here with a list of Khan Academy topics you should have down before you take physics, but there is also a really great site with free worksheets called Kuta Software. They have free worksheets for everything up to calculus, and what’s especially great is that they all have answers.

Here are the ones I would recommend you do before you take physics:

Trig Problems

Pythagorean Theorem

Inverse Trig Functions

Basic Algebra

Solving Systems of Equations

Somewhat harder systems of equations (harder than what you’ll have to do in physics)

Parabolas

Complex Fractions (These just tend to trip people up, I’ve noticed)

If you can do all these no problem, then you’re ready for basic, algebra-based physics.

If you’re going to take calculus-based physics, you might want to do these, too:

Derivatives

Higher-order derivatives

Product Rule

Quotient Rule

Chain Rule

Derivative at a value

Integration: Power Rule

Integration: Logs and Exponentials

Integration: Substitution

Depending on how in-depth the class is, there may or may not be more that you’ll need to know, calculus-wise, but the above cover the basics as far as being able to take derivatives and integrals goes.

# What to do if you’re taking calculus-based physics next year and you haven’t taken calculus yet

Every year I have a couple of students who are taking calculus-based physics at the same time as calculus. Traditionally, you take calculus first, and then  you take physics with calculus. The calculus used in these introductory physics classes is usually simple enough that it’s not impossible to take them both at the same time, but it’s definitely a challenge. I wouldn’t recommend it if you can avoid it, but if you can’t there are some things you can do over the summer that will make next year go much more smoothly:

• Recognize that you’re at a disadvantage. The worst thing possible outcome is that you end up thinking you’re bad a physics. Even if there isn’t much calculus in the course, a lot of the ideas in physics are covered in calculus. There is a lot of conceptual overlap. So, just keep in mind that everyone else who took calc last year has had a whole year to get comfortable with these ideas and to practice them. Also, you’re having to learn two challenging things at once. Not only that, but you’re having to put them together. It’s as if you just learned how to juggle and you also just learned how to ride a bike, but you’re forced to practice them both at the same time.
• Get a head start on calc. I recommend Calculus Made Easy, which is actually the book I used to help me learn calc when I was in high school. Start at the beginning and work up through chapter 8. Take your time. Focus more on the ideas than on the math.
• Work through my book Rates of Change. I feel like recommending this to you is a bit of a conflict of interest on my part, and I agonized over whether to include it in this post, but I really do think it helps make the fundamental ideas much clearer.

Lastly, once school starts, don’t hesitate to ask for help, either from your teacher, a tutor, or me.

# Summer Math Review Puzzle: Algebra I #1

One of my students asked if I would write her some math puzzles to help her review Algebra I over the summer before she goes into Algebra II. I thought I’d post them here, too! Hope you like it!

# What I wish someone had told me when I was just starting out as a physics major

(And a girl. Not that I was starting out as a girl. I had been a girl my whole life, but I was just starting out being a girl physics major.)

It’s not supposed to be easy

I had this idea that if you were smart things were easy. You either knew something or you didn’t. You either got it or you didn’t. My first few physics classes reinforced this. I had already taken high school physics, so the intro classes weren’t so bad. Then I got to my first real physics class, Mathematical Physics, taught by an old Russian professor. I assumed it would be easy. Then the first midterm came. Three problems. I had no idea what to do for any of them. I had no idea how they even related to anything we’d done in the class so far.

Suddenly I felt like I was out of my depth. I went to go talk to the professor. I had skipped class a couple of times and I think he knew that because he wasn’t that keen on helping me. What he said was “Maybe you’re just not going to get it.” After that I panicked. I felt deep down that I just wasn’t smart enough to study physics, even though it was something I had loved since I was a kid. I thought maybe I had just hit the limit of my natural intelligence.

I hadn’t. I had simply hit the limit of the place where I had learned things before. In my experience, 90% of the things that look like talent are actually practice. When you see someone with a perfect golf swing or a genius math brain, you’re only seeing the result, you’re not seeing the great coach they had, the hours they spent practicing, or all the times when they were a little girl sitting in the passenger seat of their dad’s truck talking about the differences between even and odd numbers.

I didn’t know this. I thought I had just hit the ceiling of what I was capable of. I wish I could go back and tell myself that there is no ceiling. There is no capable, not capable, or highly capable.

I work with a lot of students, and I can see some of them falling into this trap. They encounter a difficult problem and they start thinking they’re dumb. They’re wrong.

It’s hard because it is made of ambiguous problems. A lot like life. You learn some tools, some equations, some concepts, and then they set you loose on problems. At first, you will have no idea what to do. This is not bad. This is a feature of real problems. (See my post on problem solving for ways of tackling these things.)

In order to be successful in physics, you have to make the mind shift from “I’m not smart enough” to “this is hard.”

It’s okay. It’s okay if it’s hard. It’s okay if you look at a problem and have no idea what to do. It’s not supposed to be easy.

Unlike in most of your previous schooling, you’re not just learning one thing at a time and being asked to do something repetitively. You’re being asked to do something crazy hard, which is, stumble around until the pieces fit together. Or maybe they won’t, and you’ll ask for help. But not from someone who will tell you that maybe you won’t ever understand. Because if you keep trying you will.

Some people pretend they know more than they do

Another thing that left me feeling stupid was how much my classmates seemed to know. For some reason, the reputation that physics has for being difficult brings out some pretty intense egos. Be careful not to assume, just because someone is talking about something fancy-sounding that you’ve never heard of, that they know more than you do.

In this same first physics class, I met several people. We studied together. They talked a lot about things in ways that made me feel like they knew what they were talking about. Again, I felt like an idiot who didn’t belong.

Then we got our grades back. I did the worst I had ever done in a class, but it was still a 3.1. Those same people who I thought knew everything got 2.5’s or worse.

Never think someone knows more than you do just because they’re speaking authoritatively. They might just be covering insecurity with arrogance.

The next quarter I started my next physics class. I was completely panicking. Sure I would fail. Sure I wasn’t good enough. I had to take mathematical physics II.

Then, randomly, I did the best two things I’ve ever done for myself in the context of my physics degree.

1. I introduced myself to some other girls in that class.

2. I went to the professor after class, told him I was worried I was unprepared and that I wouldn’t do well.

The first thing immediately diminished my anxiety. Suddenly, I had people to commiserate with, to study with, to hang out with and laugh with. It was amazing and great and so much fun and the opposite of isolating. They told me later that they were so glad I had introduced myself, because they were feeling terrible and scared, too. It’s hard to do that, because sometimes other people look arrogant when they’re scared. Or they look closed and unfriendly when they’re scared. But, if you just introduce yourself it can be amazing. Having allies is probably the most important thing when you’re trying to do something hard.

The second thing was really good for my grade. Luckily, this professor was an extremely, extremely, nice guy. He laughed a little when he heard what my previous grade was. He told me it would be ok. He gave me a book. This book was basically a more detailed version of the textbook we’d used the previous quarter. It was waaaay better. (Incidentally, I do not recommend you use any textbook by Mary Boas. She leaves out waaay too many steps when she solves math equations.)

In that class I got a 3.9, and I made some awesome friends.

At some point in your life you will (hopefully) reach a place that will feel like the limit of your ability. It is not. It is the end of the mapped territory. It is the end of the things you already know how to do. It is a place where you will need to make a large step up in order to continue. Maybe physics will be this for you, maybe it won’t. Maybe it will be easy for you. If so, that’s great, but I hope you eventually find yourself in that place where you think you might just not be good enough.

# Basic Orbital Dynamics

This explanation was written up by Caleb Pool, who very kindly gave me a bunch of awesome feedback on my Kinematics book, and then wrote this cool explanation of Orbital Dynamics that I thought you might find interesting/useful/entertaining.

# Basics of Orbital Dynamics

When we look outside, we see ground that is pretty flat.  Trees go straight up, buildings stand up, and the ground is perpendicular to those.

Even buildings look like they stand up straight.

Water looks very flat when we stand next to it.

If one side of a pond is higher than the other, the water flows until it is level again.

Gravity pulls the water equally until it looks flat from a close perspective.  But what about when we start zooming out?

The water still looks flat to us.  We zoom out some more.

And some more:

And some more:

And a lot more:

It seems the earth is round.  But we knew that.  How can all the water stay on without falling off?  Gravity pulls objects (and water) toward the center of the earth.

Although, from our perspective, things fall down and water looks flat, the bigger story is that things fall toward the earth’s center and water curves around the earth.

The earth spins about an axis passing through the poles. This rotation causes a force that is largest around the equator, and it makes the earth’s water and land bulge slightly:

This bulge makes the earth’s radius through its equator about 3,963 miles (6,378 km), when its radius through the poles is 3,950 miles (6,356 km).  A bulge that causes a 13-mile difference in radius between the poles and equator seems like a lot, and it is, but that bulge is not very big compared to the earth’s overall size.   That bulge plays an important role in many orbits.  The basics of an orbit work even if we consider the earth to just be a sphere for now.

Suppose you find an extremely high mountain that reaches above the earth’s atmosphere and you hike to the top, carrying an extremely powerful cannon.

You fire the cannon.

But that’s not far enough.  You increase the power and fire again and again:

You keep increasing the power on this magic cannon and firing away from your mountaintop.  The cannonballs fly so far that, as they fall toward the earth, the earth starts curving away underneath them.

You aren’t satisfied with just shooting cannonballs so far away they land on the opposite side of the earth, so you continue to increase the power:

And look what just happened!  The cannonball went all the way around the world!  If you step out of the way, you get to watch it zoom by really fast (remember, you had to fire it fast enough to go that far!).  What happens now?  The cannonball doesn’t just stop dead in its tracks after it goes around once; it keeps going around.  And around.

It turns out that you have just made a satellite and launched it into orbit.  Until it gets slowed down by the thin atmospheric gas at the edge of space and burns up, your cannonball will continue to fly through space, following the same path as it circles the world.  Your special cannonball now joins the moon (earth’s only natural satellite) and a host of about 40,000 man-made objects as it endlessly circles in orbit.

As gravity pulls the cannonball toward the earth, it falls, but the earth’s surface curves away just as fast, so it never gets any closer to the earth as it goes around.

Now that you are satisfied with your powerful cannon, you look at other orbiting objects to see how they behave.

Objects in orbit that are farther away from the earth go around it more slowly.  This is primarily because earth’s gravity is weaker the farther away from the earth you go.

Example:  Which is farther away from the earth?  The moon, which goes around once a month, or a GPS satellite, which goes around twice each day?

Answer:  the moon is farther away.

# The Poor Man’s Bike Speedometer

(Or, how to tell how fast you’re going based on how much mud is hitting you in the face.)

I was riding my bike in the rain yesterday, along a dirt road, listening to Let it Go from Frozen, when I realized that if I leaned forward I started getting a lot of mud in my face from water kicked up off my tire.  That water gets flung off the tire at whatever the rotational speed of the wheel is. The faster I go, the higher the water goes. Maybe I could use that to tell how fast I was going. (I have a bike speedometer, but I still haven’t installed it. That was Christmas two years ago so it might never happen.)

Then I was like, but, I’d have to estimate the size of a water droplet, and if there was dirt in it that would increase the mass, so I’d also have to make some assumptions about percent dirt content.

But, no! The height of the droplet can be calculated with conservation of energy, which means the mass drops out. As long as I’m ignoring air resistance, which is probably ok, I can pretty simply calculate my speed based on how far I have to lean forward before I start getting mud in my face.

Here’s how it works:

As I ride forward, water droplets cling to the surface of the tire. At some point, they are flung off. When they are flung off, they travel in a straight line out from the surface of the tire. They travel tangentially, and their speed is equal to the linear speed of the rotating wheel.

The linear speed of the wheel is equal to the speed that I’m bicycling at. That means that the water droplet should theoretically be travelling at the same speed at which I am bicycling.

Now all that I have to do is use conservation of energy to find that speed, given the height.

There are two types of energy involved, linear kinetic and gravitational potential. Here are the equations:

Here, m is the mass of the object, v is its velocity, h is its height above the surface of the earth in meters, and g is the acceleration due to gravity (approximately 9.8.)

The idea here is that energy is never created or destroyed. It just changes forms. In this case, it changes from kinetic energy to potential energy.

(I’m making several simplifications here. I’m assuming no air resistance, and no energy lost to deformation of the water droplet.)

Ok, here’s the calculation:

This equation at the end you can use to find your velocity. Just plug in the height your head is above the wheel when you start getting mud in your face.

Now for some numbers.

My usual head height is 3 feet, 3 inches above the midline of my wheel (I’m making another simplification, assuming that my head is directly vertical above the back edge of the wheel:

I’ll need to convert that to meters. For that I’ll just use google’s handy conversion feature.

Apparently 3’3″ is .9906 m

Plug this in to my equation:

And convert to miles per hour:

Here’s a table of head heights versus bike speed:

The cool thing is that it doesn’t matter how big your tire is, or how big the water droplets are, or how much dirt the droplets have in them. All that matters is how fast you’re going and how high above the wheel your face is.

There is probably a cutoff point, where if you’re going slower than that the water isn’t lifted at all, and of course if you’re going too fast you won’t be tall enough to reach the height of the water. Also, you can’t have fenders on your bike.

But, that’s how you can tell how fast you’re going based on how much mud you get in the face!

# Electric Circuits Education Resources

For learning the basics, hands on activities can be great.

Here is one of the coolest things I’ve ever seen for teaching electric circuits:

Squishy Circuits

Of course, the Phet Simulations are also great for inciting interest and sparking curiosity.

Now, for some resources for the more advanced:

Electrodynamics (More about E and B fields than Circuits, but great.)

The Art of Electronics (Basically the bible for electric circuits.)

Enjoy! And, if you’re somewhere in the middle, or would like a tutorial on problem solving, and the basic laws of circuits like kirchhoff’s and ohm’s laws, you can check out my book.  It’s been either #1 or #2  in electricity principles, and electromagnetism, and AP test guides for months now. If you do check it out I would very much appreciate an honest review (whether you like it or hate it.)

# Fun Practical Applications of Physics

Here’s a fun video of some physics-based tricks. (Called 10 Amazing Bets you will Always Win, by Quirkology)

A couple of them (moving liquid from a bottle to a glass and separating two glasses) use air pressure creatively.

Physics can also be used to make some super-cool art.  Have you seen the Cloud Gate in Chicago?  On the outside it’s an amazing panorama of the city skyline, on the inside it’s a crazy multidimensional universe. That’s the magic of curved mirrors.

# 6 Tricks for Solving Physics Problems

In a physics problem, there is usually more information than you can comfortably keep in your head at once.  You need a good diagram, a good list, a good way of keeping track of information. The diagram/list/table/picture should represent what you know up until this point.

If you’re having trouble solving a problem, ask yourself this question: “Is everything I know represented in my diagram?”

It’s important to put even seemingly self-evident stuff into your diagram, because sometimes what’s holding you back is that you haven’t seen all the implications of what you know is true.  Put the obvious stuff in your diagrams because then you’ll see the less-obvious implications.

Problem solving is a process of taking one small step at a time.  With a really good problem, you probably won’t be able to see the solution, or even how to get to the solution, until you are nearly there.  Instead, you just keep asking yourself “what can I try now?”

Solve a Simpler Problem

A good example of this is what xkcd calls “The Hardest Logic Puzzle in the World.” I won’t summarize it here- you can go read it and come back.

Back? Ok.  This problem is really really hard.  First, do step one and figure out a good way to draw the problem, maybe just with L’s for blue-eyed people and R’s for Brown-eyed people.

Then, step two, simplify the problem.  Thinking about what a hundred people know (about themselves, and about every other person) is a bit of a stretch. Instead, try this problem with just one islander. Seriously. Just one.

If the guru says I see at least one blue eyed person, then that one guy knows he has blue eyes, and he leaves.  Yay! That was easier. At this point, we’ve basically solved the problem.  All we have to do is figure out what happens when there are more people.

Try it with two islanders.  Then three.  Then four. Then extrapolate to 50 once you see the pattern.  Easy! Jusy kidding, it’s not easy.  But figuring out a simpler problem to solve makes it waaay easier.  I won’t deprive you of the logical joy of this problem by telling you the answer.

Try Something

You could probably simulate being tutored by me pretty well just by asking yourself “what is one thing I can figure out at this point?”

I love asking students this, because it shows them that they already have all the knowledge they need to solve a problem, and the capability to do so, they just need the willingness to try.  (Some students of course need some explanations, but you’d be surprised how often students who come to me failing physics and feeling completely confused actually do just great after a couple of meetings and all I said the whole time was “what’s one thing you could figure out?” I exaggerate a little, but not all that much.)

Physics is hard not because there is a lot you have to learn, but because applying the small amount of learned information takes courage.  It takes the courage to try something and see what happens and trust yourself that if you try enough things you’ll eventually find your way to the answer.

The problems are often so complex that it’s impossible, until you become an expert, to map out the path to the solution.  Instead, you have to just figure out one thing, then another, then another, until a way to find what you’re looking for jumps out at you.

Can you solve for the time? Can you find the total force?  Can you find the mass?

(Mass by the way, is often something that drops out.  If you find yourself confused because “they don’t give you the mass” then just use m for mass and be happily surprised when it magically cancels out a few steps later.)

Just do whatever you can do, write it all down, be as organized as possible, and you will find the answer… eventually… most of the time.

Use Difference Reduction

One tried and true method for solving problems in any context, including physics, is called difference reduction.  In this method, you simply look at what you have and look at what you want and ask yourself, what’s one small change that will make what I have look more like what I want?

I’m starting to feel a little bit like a life coach now, but hey, real life problems are problems too.

Sometimes to move forward you must go back

Have you heard the riddle about the farmer crossing the river with the fox, the goose, and the bag of beans?  There are tons of versions of this riddle, but here’s one:

A farmer has a fox, a goose, and a bag of beans and he wants to cross a river. His boat will only hold one item at a time in addition to the farmer.  If he leaves the fox with the goose the fox will eat the goose, and if he leaves the goose with the beans the goose will eat the beans.  How can he do this?

I’m going to give you a partial spoiler here in about two lines, so if you want to try this on your own now is a good time.

*

*

*

*

The key to answering this problem is that you have to bring the goose back to the wrong side of the river at one point.  The reason this riddle is hard is that it involves moving away from your goal state.  You already have the goose on the right side, why would you bring it back?

But, in many contexts, taking a step back is the best way forward. Maybe you move home after college.  This feels like the worst step back in the world, but you save money and buy a house and a car much sooner than you would have otherwise.  Maybe you’ve been working on a physics problem for three hours and you feel like you’re almost there but you just can’t get the last little bit.  This might mean that you’ve actually gone down the wrong path and if you keep on doing exactly what you’ve been doing you’ll never get there and what you really need to do is start over.  Before you start over, though, you should:

Take a Break

There is this thing called neuronal activation.  Certain neurons in your brain light up, and they have their own associations with them, which makes you more likely to think of things in one way rather than in another.  Sometimes if you’re having trouble with a problem it’s because you’re thinking of it in one specific way that isn’t letting  you see the route to the answer.

If you take a break you can come back fresh, with different neural pathways activated, which means you might just see the answer right away.  (There is some disagreement about whether this is because of the “fresh perspective” or because your subconscious was working on the problem without you.  Either way it works.)

# Real World Physics Applications – Thermodynamics

There’s something kind of awesome that happens when you realize that things you’ve learned in the classroom might actually have real-world applications.  Like, when you go to a foreign country and can  sort of communicate with the locals because you took Spanish 101.

I was watching Shark Tank recently, and these guys came on with this thing they invented.  I don’t know anything for sure about how it works, but I could make some educated guesses, and I think that looking at their invention is a really interesting way to introduce thermodynamics.

I couldn’t find the clip of them on shark tank, but here’s their kickstarter campaign video, which according to youtube raised \$306k in 37 days!

Anyway, it’s a pretty genius invention and I kind of want one. They’re these big metal things shaped like coffee beans and you put them in your coffee.  They immediately cool it to the perfect (still hot) temperature for drinking, and then keep your coffee at that temperature for several hours.

Being a coffee drinker, extreme appreciater, addict, myself, I was like “wow, that’s genius.”

Here’s my guess at how it works: The inside is some metal that has its melting point at a good temperature for coffee.  When you pour coffee over it, the excess heat goes into melting the metal. (This phase change absorbs energy, (called Joules, hence the name Coffee Joulies.)

Then, as the coffee starts to be cooled by the outside air, the metal inside starts to phase change back, solidifying and transferring heat back to your coffee.  (Phase changes can absorb tons of energy, and they will stay at a constant temperature while they do so.)

So simple!

We could even guess at what metal or material might be inside.  It would be something with a melting point around the perfect coffee drinking temperature.

The National Coffee Association says this temperature is 180-185 Fahrenheit, which is 82-85 Celsius. But, drinks.seriouseats.com says 110-120 is ideal, which is 43-49 C.  Coffee Joulies says they keep your coffee at exactly 140, or 60 C.

Then we can check the melting points on the periodic table. Sodium (Na) is 98 C and Potassium (K) is 63 C, Rubidium is 39 C.  Potassium is basically right on, so it could conceivably be made of that.

I just checked their website for more clues. They call it PCM (for potassium? Maybe it’s a mixture of potassium and something else.) Other than that they don’t say what it is, only that it’s natural, edible, and already found in food (potassium again?)

Anyway, fun mystery, and an awesome invention!  And, a really awesome example of how you can use physics to invent useful things.