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Physics Calculator

Get detailed solutions to your math problems with our physics step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of physics. This solution was automatically generated by our smart calculator:

What do we already know? We know the values for acceleration ($a$), velocity ($v$), distance ($y$), height ($y_0$) and want to calculate the value of velocity ($v_0$)

According to the initial data we have about the problem, the following formula would be the most useful to find the unknown ($v_0$) that we are looking for. We need to solve the equation below for $v_0$

We substitute the data of the problem in the formula and proceed to simplify the equation

Multiply $-2$ times $9.81$

Multiply $-19.62$ times $3.2$

Calculate the power $0^2$

Rearrange the equation

We need to isolate the dependent variable $v_0$, we can do that by simultaneously subtracting $-62.784$ from both sides of the equation

 Intermediate steps

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$

Divide $1$ by $2$

Simplify $\sqrt{v_0^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $0.5$

Multiply $2$ times $0.5$

Calculate the power $\sqrt{62.784}$

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1.7 Solving Problems in Physics

Learning objectives.

By the end of this section, you will be able to:

• Describe the process for developing a problem-solving strategy.
• Explain how to find the numerical solution to a problem.
• Summarize the process for assessing the significance of the numerical solution to a problem.

Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.

As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

• Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
• Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch can be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
• Identify exactly what needs to be determined in the problem (identify the unknowns) . In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
• Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

• Check your units. If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
• Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
• Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these that science progresses.

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Problem Solving

Linear algebra can be used to solve large ohmic circuits. This page explains how to define the circuit problem in terms of a linear system that can be solved with computational methods like matlab or python.

• 1.1 Incidence Matrix
• 1.2 Loop Rule (Kirchhoff's voltage law)
• 1.3 Kirchhoff's Current Law
• 1.4 Solving Problems
• 3.1 Further reading
• 3.2 External links
• 4 References

The Main Idea

We can describe an electrical circuit as a graph where the nodes represent specific points and the edges represent the components connecting them. E.g:

Incidence Matrix

The incidence matrix, A, is a matrix of dimensions m x n, where m is the number of edges and n is the number of nodes. If x is the vector that contains all the nodes in order, A satisfies: [math]\displaystyle{ A\vec{x}_{edge} = x_{end} - x_{start} }[/math] Thus, the resulting matrix will have zeros in every element of each row, except for the start and the end nodes, which will be -1 and 1 respectively:

Here, [math]\displaystyle{ x_{1}, x_{2}, x_{3} }[/math] and [math]\displaystyle{ x_{4} }[/math] represent the voltages at the nodes. Then [math]\displaystyle{ A\vec{x}_{edge} = x_{end} - x_{start} }[/math] gives the voltage differences [math]\displaystyle{ ∆V }[/math] .

Loop Rule (Kirchhoff's voltage law)

Using this creative way of transforming a circuit to a graph and getting the incidence matrix from it allows us to do some interesting things. Kirchhoff's loop rule is a principle of conservation of energy that implies that the directed sum of the voltage differences around any closed network is zero.

This means that the components of [math]\displaystyle{ A\vec{x} = b }[/math] add to zero around every loop. Moreover, the voltage law can decide whether b is in the column space of A.

We know from Ohm's law that [math]\displaystyle{ V = I * R }[/math] . Then, in the previous graph for example, we would have that [math]\displaystyle{ V_{edge1} = I * R = (direction) * R_{1} = 1 * R_{1} }[/math] .

Kirchhoff's Current Law

We know from Kirchhoff's law that the sum of the currents in each node is zero. This means that the current flowing into the node is the same as the current flowing out of the node. Therefore the multiplication of each column of A by the current vector, y, must be equal to zero:

[math]\displaystyle{ A^{T}\vec{y} = 0 }[/math]

where [math]\displaystyle{ y_{i} }[/math] is the current passing through edge [math]\displaystyle{ i }[/math] .

If a circuit has a source like a battery, we can treat the entire circuit as a node. The current in and out of every node within the circuit will be zero except for the nodes connected to the positive and negative ends of the battery. For the positive end the net current, [math]\displaystyle{ S }[/math] , will be positive and for the negative end the net current will be negative. Thus we get:

[math]\displaystyle{ A^{T}\vec{y} = f }[/math]

where [math]\displaystyle{ f_{i} = 0 }[/math] if [math]\displaystyle{ i }[/math] is not one of the nodes connected to the battery, [math]\displaystyle{ f_{i} = S }[/math] if [math]\displaystyle{ i }[/math] is connected to the positive end and [math]\displaystyle{ f_{i} = -S }[/math] if [math]\displaystyle{ i }[/math] is connected to the negative end.

Solving Problems

From Ohm's law we have [math]\displaystyle{ V = I * R }[/math] . Thus if we define the conductance diagonal matrix [math]\displaystyle{ C }[/math] where [math]\displaystyle{ C_{i,i} = \frac{1}{R_{i}} }[/math] , [math]\displaystyle{ C_{i,j} = 0 }[/math] , Ohm's law becomes:

[math]\displaystyle{ \vec{y} = -CA\vec{x} }[/math]

where [math]\displaystyle{ y_{i} }[/math] is the current through edge [math]\displaystyle{ i }[/math] and [math]\displaystyle{ x_{j} }[/math] is the voltage at node [math]\displaystyle{ j }[/math] . Substituting for y in Kirchhoff's law, we get:

[math]\displaystyle{ -A^{T}CA\vec{x} = f }[/math]

We can now use this system of matrices to solve circuit problems.

It is important to note that for the exercises posted above, the solutions for those problems should be easy to compute given the number of nodes and connections. Nevertheless, bigger and more complex problems would benefit massively from the methods of linear algebra, as they could be computed in this standard way.

Suppose we have the following circuit (along with its graph and matrix representation):

Solving the system, we get:

Notice that [math]\displaystyle{ C }[/math] is the identity, thus:

Solving for x, we get the voltages:

Finally, we can find the currents using [math]\displaystyle{ \vec{y} = -CA\vec{x} }[/math] :

Further reading

• Linear Algebra applications in image compression: https://math.mit.edu/~gs/linearalgebra/linearalgebra5_7-1.pdf

External links

Gilbert Strang's "Equilibrium Equations and Minimum Principles": http://cns-web.bu.edu/~eric/EC500/attachments/ON(2d)LINE(20)READINGS/strang.pdf

• Strang, G. (2016), Introduction to Linear Algebra , Wellesley-Cambridge Press, 5th edition, isbn:978-0-9802327-7-6
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How to Solve Any Physics Problem

Last Updated: July 21, 2023 Fact Checked

This article was co-authored by Sean Alexander, MS . Sean Alexander is an Academic Tutor specializing in teaching mathematics and physics. Sean is the Owner of Alexander Tutoring, an academic tutoring business that provides personalized studying sessions focused on mathematics and physics. With over 15 years of experience, Sean has worked as a physics and math instructor and tutor for Stanford University, San Francisco State University, and Stanbridge Academy. He holds a BS in Physics from the University of California, Santa Barbara and an MS in Theoretical Physics from San Francisco State University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 328,864 times.

Baffled as to where to begin with a physics problem? There is a very simply and logical flow process to solving any physics problem.

• Ask yourself if your answers make sense. If the numbers look absurd (for example, you get that a rock dropped off a 50-meter cliff moves with the speed of only 0.00965 meters per second when it hits the ground), you made a mistake somewhere.
• Don't forget to include the units into your answers, and always keep track of them. So, if you are solving for velocity and get your answer in seconds, that is a sign that something went wrong, because it should be in meters per second.
• Plug your answers back into the original equations to make sure you get the same number on both sides.

Community Q&A

• Many people report that if they leave a problem for a while and come back to it later, they find they have a new perspective on it and can sometimes see an easy way to the answer that they did not notice before. Thanks Helpful 249 Not Helpful 48
• Try to understand the problem first. Thanks Helpful 186 Not Helpful 51
• Remember, the physics part of the problem is figuring out what you are solving for, drawing the diagram, and remembering the formulae. The rest is just use of algebra, trigonometry, and/or calculus, depending on the difficulty of your course. Thanks Helpful 115 Not Helpful 34

• Physics is not easy to grasp for many people, so do not get bent out of shape over a problem. Thanks Helpful 100 Not Helpful 25
• If an instructor tells you to draw a free body diagram, be sure that that is exactly what you draw. Thanks Helpful 89 Not Helpful 24

Things You'll Need

• A Writing Utensil (preferably a pencil or erasable pen of sorts)
• Calculator with all the functions you need for your exam
• An understanding of the equations needed to solve the problems. Or a list of them will suffice if you are just trying to get through the course alive.

You Might Also Like

Expert Interview

Thanks for reading our article! If you’d like to learn more about teaching, check out our in-depth interview with Sean Alexander, MS .

• ↑ https://iopscience.iop.org/article/10.1088/1361-6404/aa9038
• ↑ https://physics.wvu.edu/files/d/ce78505d-1426-4d68-8bb2-128d8aac6b1b/expertapproachtosolvingphysicsproblems.pdf
• ↑ https://www.brighthubeducation.com/science-homework-help/42596-tips-to-choosing-the-correct-physics-formula/

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Scientists use generative AI to answer complex questions in physics

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When water freezes, it transitions from a liquid phase to a solid phase, resulting in a drastic change in properties like density and volume. Phase transitions in water are so common most of us probably don’t even think about them, but phase transitions in novel materials or complex physical systems are an important area of study.

To fully understand these systems, scientists must be able to recognize phases and detect the transitions between. But how to quantify phase changes in an unknown system is often unclear, especially when data are scarce.

Researchers from MIT and the University of Basel in Switzerland applied generative artificial intelligence models to this problem, developing a new machine-learning framework that can automatically map out phase diagrams for novel physical systems.

Their physics-informed machine-learning approach is more efficient than laborious, manual techniques which rely on theoretical expertise. Importantly, because their approach leverages generative models, it does not require huge, labeled training datasets used in other machine-learning techniques.

Such a framework could help scientists investigate the thermodynamic properties of novel materials or detect entanglement in quantum systems, for instance. Ultimately, this technique could make it possible for scientists to discover unknown phases of matter autonomously.

“If you have a new system with fully unknown properties, how would you choose which observable quantity to study? The hope, at least with data-driven tools, is that you could scan large new systems in an automated way, and it will point you to important changes in the system. This might be a tool in the pipeline of automated scientific discovery of new, exotic properties of phases,” says Frank Schäfer, a postdoc in the Julia Lab in the Computer Science and Artificial Intelligence Laboratory (CSAIL) and co-author of a paper on this approach.

Joining Schäfer on the paper are first author Julian Arnold, a graduate student at the University of Basel; Alan Edelman, applied mathematics professor in the Department of Mathematics and leader of the Julia Lab; and senior author Christoph Bruder, professor in the Department of Physics at the University of Basel. The research is published today in Physical Review Letters.

Detecting phase transitions using AI

While water transitioning to ice might be among the most obvious examples of a phase change, more exotic phase changes, like when a material transitions from being a normal conductor to a superconductor, are of keen interest to scientists.

These transitions can be detected by identifying an “order parameter,” a quantity that is important and expected to change. For instance, water freezes and transitions to a solid phase (ice) when its temperature drops below 0 degrees Celsius. In this case, an appropriate order parameter could be defined in terms of the proportion of water molecules that are part of the crystalline lattice versus those that remain in a disordered state.

In the past, researchers have relied on physics expertise to build phase diagrams manually, drawing on theoretical understanding to know which order parameters are important. Not only is this tedious for complex systems, and perhaps impossible for unknown systems with new behaviors, but it also introduces human bias into the solution.

More recently, researchers have begun using machine learning to build discriminative classifiers that can solve this task by learning to classify a measurement statistic as coming from a particular phase of the physical system, the same way such models classify an image as a cat or dog.

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner.

The Julia Programming Language , a popular language for scientific computing that is also used in MIT’s introductory linear algebra classes, offers many tools that make it invaluable for constructing such generative models, Schäfer adds.

Generative models, like those that underlie ChatGPT and Dall-E, typically work by estimating the probability distribution of some data, which they use to generate new data points that fit the distribution (such as new cat images that are similar to existing cat images).

However, when simulations of a physical system using tried-and-true scientific techniques are available, researchers get a model of its probability distribution for free. This distribution describes the measurement statistics of the physical system.

A more knowledgeable model

The MIT team’s insight is that this probability distribution also defines a generative model upon which a classifier can be constructed. They plug the generative model into standard statistical formulas to directly construct a classifier instead of learning it from samples, as was done with discriminative approaches.

“This is a really nice way of incorporating something you know about your physical system deep inside your machine-learning scheme. It goes far beyond just performing feature engineering on your data samples or simple inductive biases,” Schäfer says.

This generative classifier can determine what phase the system is in given some parameter, like temperature or pressure. And because the researchers directly approximate the probability distributions underlying measurements from the physical system, the classifier has system knowledge.

This enables their method to perform better than other machine-learning techniques. And because it can work automatically without the need for extensive training, their approach significantly enhances the computational efficiency of identifying phase transitions.

At the end of the day, similar to how one might ask ChatGPT to solve a math problem, the researchers can ask the generative classifier questions like “does this sample belong to phase I or phase II?” or “was this sample generated at high temperature or low temperature?”

Scientists could also use this approach to solve different binary classification tasks in physical systems, possibly to detect entanglement in quantum systems (Is the state entangled or not?) or determine whether theory A or B is best suited to solve a particular problem. They could also use this approach to better understand and improve large language models like ChatGPT by identifying how certain parameters should be tuned so the chatbot gives the best outputs.

In the future, the researchers also want to study theoretical guarantees regarding how many measurements they would need to effectively detect phase transitions and estimate the amount of computation that would require.

This work was funded, in part, by the Swiss National Science Foundation, the MIT-Switzerland Lockheed Martin Seed Fund, and MIT International Science and Technology Initiatives.

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Researchers at MIT and elsewhere have developed a new machine-learning model capable of “predicting a physical system’s phase or state,” report Kyle Wiggers and Devin Coldewey for TechCrunch . 

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Title: physics informed cell representations for variational formulation of multiscale problems.

Abstract: With the rapid advancement of graphical processing units, Physics-Informed Neural Networks (PINNs) are emerging as a promising tool for solving partial differential equations (PDEs). However, PINNs are not well suited for solving PDEs with multiscale features, particularly suffering from slow convergence and poor accuracy. To address this limitation of PINNs, this article proposes physics-informed cell representations for resolving multiscale Poisson problems using a model architecture consisting of multilevel multiresolution grids coupled with a multilayer perceptron (MLP). The grid parameters (i.e., the level-dependent feature vectors) and the MLP parameters (i.e., the weights and biases) are determined using gradient-descent based optimization. The variational (weak) form based loss function accelerates computation by allowing the linear interpolation of feature vectors within grid cells. This cell-based MLP model also facilitates the use of a decoupled training scheme for Dirichlet boundary conditions and a parameter-sharing scheme for periodic boundary conditions, delivering superior accuracy compared to conventional PINNs. Furthermore, the numerical examples highlight improved speed and accuracy in solving PDEs with nonlinear or high-frequency boundary conditions and provide insights into hyperparameter selection. In essence, by cell-based MLP model along with the parallel tiny-cuda-nn library, our implementation improves convergence speed and numerical accuracy.

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1.8: Solving Problems in Physics

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Learning Objectives

• Describe the process for developing a problem-solving strategy.
• Explain how to find the numerical solution to a problem.
• Summarize the process for assessing the significance of the numerical solution to a problem.

Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.

As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

• Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
• Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
• Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
• Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

• Check your units . If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
• Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
• Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these science that progresses.

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Oxford geoscience professor myles allen on solving the problem of climate change.

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This week I had the pleasure of meeting Myles Allen . He is Head of Atmospheric, Oceanic and Planetary Physics in the Department of Physics, University of Oxford, and Professor of Geoscience in the School of Geography and Environment. He’s been studying how human and natural influences contribute to climate change since the early 1990s, served on the UN Intergovernmental Panel on Climate Change (IPCC) for its 3rd, 4th and 5th Assessments, and was a Coordinating Lead Author for its special report on 'the impacts of global warming of 1.5 °C above pre-industrial levels,’ and has been dubbed by the BBC “ the physicist behind net zero ”. In short, his scientific credentials regarding climate change are superb.

Professor Myles Allen Delivering The Gresham Lecture on May 21, 2024

But unlike many science professors, Professor Allen is bravely wading into the world of climate policy. The timing of my meeting with Professor Allen was fortuitous. It was the day before he gave a lecture on May 21, 2024, in his capacity as the Frank Jackson Professor of the Environment as part of the annual series of lectures hosted by Gresham College in the City of London. Gresham (b. 1519, d. 1579) was an English merchant, financier, and founder of the Royal Exchange. He started these lectures to ensure the emerging merchant classes were culturally enriched, as well as being versed in the latest science and technology.

The title of Professor Allen’s lecture is “ A Just and Inclusive Net Zero .” You can also view it online , which I highly recommend. The lecture is both global in its ideas and local in terms of the climate change debate taking place in the UK. I am less familiar with this than I am in the United States, and it was through this lens that I read and then listened to his lecture. My overall impression is how pragmatic and non-ideological it is. Professor Allen is no more an apologist for the fossil fuel industry than he is a supporter of the fossil fuel haters. Neither group brings value to the discussion of how to address climate change. Professor Allen does.

Sir Thomas Gresham

While I can’t do justice to his full lecture in this brief summary, here are four points that leaped out to me.

The first is he notes that “in the Paris-aligned scenarios of the IPCC, we are still using fossil fuels, at around one-quarter of the current rate, in 2100, long after the date of net zero.” Today’s world population of eight billion people will have grown to nearly 10 billion, with hopefully a larger percentage leading lives closer to what is enjoyed in the developed world. That means a LOT of fossil fuels. Those who harbor the fantasy that we still have time to achieve our climate goals simply by phasing out fossil fuel use need to get real. “One of the most dangerous myths is that achieving net zero is actually going to be really cheap because carbon-free, instantly-dispatchable energy” will soon take care of all of our energy needs. Nope. Even with the best of battery technologies it will be intermittent, and we’ll need natural gas and nuclear for the baseload.

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People also need to be realistic about some of the barriers to reducing fossil fuel use as rapidly as possible, with permitting being the number one in my mind. Environmentalists protesting transmission lines for bringing renewable power into the grid are part of the problem, not part of the solution. A great local example for me is the Northern Pass project for bringing cheap hydro power from Canada to Massachusetts, which I have written about with John Skjervem, the CIO of Utah Retirement Systems. Permitting extends to domestic mining of the cobalt and rare earths for battery and renewable energy technologies in order to ensure energy security and not being dependent on places like China, Russia, and the Democratic Republic of Congo for getting them.

Second, he calls out what he calls the “climate establishment” filled with those who regard themselves as elite experts who know better than the average citizen what needs to be done, such as “the unelected technocrats like the [UK] Climate Change Committee , the Science-Based Targets Initiative , or the Climate Action Tracker. ” Their perceived arrogance (whether it is real or not doesn’t matter) and disregard for which climate policies will be acceptable to those who will be subject to them is, again, part of the problem, not part of the solution. You can add to that bureaucrats in Brussels. As one example, he drolly quotes the Carbon Border Adjustment Mechanism, “or CBAM, is, in a nutshell, the European Union deciding that it has the right to impose punitive tariffs on imports from countries whose climate policies a team of bureaucrats in Brussels have decided aren’t good enough. I recently heard a talk from one such bureaucrat, I’m sure a very well-intentioned and intelligent chap, in which he said ‘of course, the CBAM is not neo-colonialist.’ If you have to assure people your policy is not neo-colonialist, you have a problem.”

Barnards Inn Hall, Gresham College, Where The Gresham Lecture Is Delivered

Professor Allen rightly suggests that these elite experts should “talking to people who wouldn’t normally show up in their social networks, like populist talk-show hosts, livestock farmers – and the executives of fossil fuel companies.” To that I’d like to add conservatives who are dedicating their lives to addressing climate change. On New Year’s Eve of 2023 I wrote about them and keep finding more. The “Eco-Right” has a very important role to play, and the climate establishment needs to show a little humility (yes, conservatives can have some very good ideas!) and be less self-righteous and start talking to them.

Third, while the “climate establishment” likes to talk about a Just Transition, it largely ignores what is just for the average citizen. Not everyone can go out and replace gas with heat pumps and buy an electric car. Yes, we need to think about a Just Transition for emerging markets, but in the U.S., we also need to think about what this means for low and middle income people in both red and blue states. Professor Allen also provocatively suggests “justice for the fossil fuel industry” which extends beyond “protecting workers in carbon intensive industries, or the interests of new fossil fuel producers, in ways that that just happens, surprise, surprise, to suit the fossil fuel industry itself rather well” to the shareholders of fossil fuel companies and people who benefit from their products—which is pretty much every single one of us.

Justice for the fossil fuel industry means acknowledging its right and need to exist but also holding it accountable for the carbon it produces. As Professor Allen notes, “The greatest climate injustice of all, to my mind, is the fact that the most profitable industry the world has ever known is entirely dependent on selling a product that is causing a very serious problem and no one is even asking them to fix it.” So how to fix it? Here are a few ways that spring to my mind of how not to fix it: (1) divesting from fossil fuel stocks in the naïve belief this will keep them from producing their product (but fine to do so if you don’t believe in the long-term value proposition or you’re just not ethically comfortable holding these stocks), (2) yelling at banks who provide them financing, (3) demanding that fossil fuel companies have unrealistic plans for reducing their investments and production, (4) urging them to get into the renewable energy business (their business model and capabilities don’t lend themselves to this), (5) opposing carbon capture storage technologies using the argument they just prolong the life of the fossil fuel industry, (6) thinking that reporting on Scope 3 emissions will somehow reduce the demand of their customers, and (7) filing useless shareholder proposals using the language of “value creation” to mask a basic hatred of the industry (which all the haters depend on).

Professor Myles Allen: “They just told me to write some physics on the board”

Which gets me to my fourth and last point. What is the solution? “There really is only one way to stop fossil fuels from causing global warming before the world stops using fossil fuels: we have to capture the carbon dioxide they generate and dispose of it, permanently, back underground.” The “underground” part is important. Turning fossil carbon into trees, water, and topsoil only delays its release into the atmosphere. The fossil fuel industry should take responsibility for doing this. It can start modestly, say one percent of the carbon it produces and building up to 100% by 2050. Let’s call this number the “geologically stored fraction.” This is what investors should be focused on, not Scope 3 emissions (although Scope 1 and 2 are fair game). Making this happen will require a mix of regulatory and market forces.

Can this be done? Absolutely. While fossil fuel companies don’t know wind and solar, they for sure know geological carbon management. The industry has a history of innovation and deep technological and engineering expertise for getting fossil fuels out of the ground, which often involving injecting stuff back into the ground. No question in my mind that it can figure out effective and cost efficient ways of putting carbon dioxide back underground at the scale required. In the beginning, the costs of doing so may be so small they just become a cost of doing business and preserving its needed license to operate. Ultimately, the costs will be borne by both the industry and its customers, which includes all of us. But if we want less carbon in the atmosphere but still want to enjoy the benefits of generating it, we must all be willing to pay our fair share for getting rid of it.

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson