In the realm of physics, understanding the concept of momentum is crucial. The equation for momentum is expressed as ( p = mv ), where ( p ) represents momentum, ( m ) is mass, and ( v ) is velocity. Essentially, momentum is the product of an object’s mass and its velocity. This relationship can be visualized graphically, with momentum on one axis and velocity on the other, where the mass determines the slope of the line.
When we calculate the area under this graph, it resembles the area of a triangle, which is calculated as half the base times the height. This calculation holds true regardless of the slope, resulting in an area of ( frac{1}{2} times P times V ). By substituting ( P = mv ), we derive the area as ( frac{1}{2} times M times V times V ) or ( frac{1}{2} mv^2 ). This equation might be familiar as it represents the kinetic energy of a moving object. The mathematical process we used to connect these equations is known as a derivation in physics.
Derivations are a method of developing new insights in physics by combining existing knowledge in innovative ways. At the university level, derivations typically involve algebra and calculus. Algebra involves rearranging or combining equations, as demonstrated when we substituted ( P = mv ). Calculus, on the other hand, consists of two main operations: integration and differentiation. Integration helps find areas under curves, while differentiation is used to determine the gradients of curves. Calculus is essential in physics as it describes how quantities change relative to one another, making it an invaluable tool.
In our example, we didn’t need calculus because the graph was a straight line, allowing us to use basic geometry to find the area. Calculating the gradient of a straight line is straightforward; it’s simply the height divided by the width of the triangle, which corresponds to the mass ( M ). However, when dealing with curved lines, such as the plot of kinetic energy with the ( V^2 ) term, calculus becomes necessary.
Integration involves approximating the area under a curve using a series of rectangles. Each rectangle’s area is expressed as ( frac{1}{2} MV^2 ), representing the height times a small width, denoted as ( Delta V ). The total area is the sum of these rectangles. As the width of these rectangles decreases, the approximation becomes more accurate, reaching perfection when the width approaches zero. This sum is then expressed as an integral.
Differentiation, conversely, is the process of finding the gradient using tiny triangles at each point. The gradient is expressed as ( frac{Delta left(frac{1}{2} MV^2right)}{Delta V} ). As these triangles become infinitely small, the approximation becomes exact, resulting in a differential equation. Solving this equation brings us back to ( MV ), illustrating that differentiation is the inverse of integration.
While this overview provides a glimpse into calculus, further exploration is recommended. The 3Blue1Brown channel offers an excellent playlist on calculus, delving into greater detail. For physicists, calculus is as essential as a wrench is to a plumber. In school, students often memorize equations without understanding their origins, but learning derivations can be incredibly rewarding.
During my university studies, understanding the derivations of equations and their interconnections was a significant achievement. If you have the opportunity to learn calculus, I highly recommend it, even for personal interest. It is applicable in many fields, and understanding how to derive new physics is a compelling reason to learn.
While algebra and calculus are foundational for physicists, other mathematical tools may be necessary depending on your field of study. These include matrix algebra, vectors, and tensors.
If you’re interested in learning more about derivations and applying them to new situations, consider exploring Brilliant.org. This platform offers problem-solving exercises that encourage careful thought and help build intuition about how different concepts relate. This approach facilitates applying knowledge in novel situations rather than merely memorizing equations.
For those interested, Brilliant.org offers a 20% discount on the annual premium membership for the first 200 sign-ups, unlocking all their content. Thank you for engaging with this article! Stay tuned for more content, as there are exciting projects on the horizon.
Create a graph with momentum on one axis and velocity on the other. Use different masses to plot multiple lines. Analyze how the slope changes with mass and discuss the implications of these changes on momentum.
Work in groups to derive the kinetic energy formula from the momentum equation. Use algebraic manipulation and discuss each step to understand the connection between momentum and kinetic energy.
Engage in exercises that involve integrating and differentiating simple functions. Use these exercises to understand how calculus is applied to find areas under curves and gradients of curves in physics.
Research and present a real-world physics problem where calculus is essential. Explain how integration and differentiation are used to solve the problem and the significance of these solutions in practical applications.
Explore other mathematical tools like matrix algebra, vectors, and tensors. Discuss how these tools complement calculus in solving complex physics problems and their applications in various fields of study.
Here’s a sanitized version of the provided YouTube transcript:
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In physics, the equation for momentum is ( p = mv ). This means that if you have a certain mass moving at a certain velocity, its momentum is equal to the mass times the velocity. Like any equation, you can represent this as a graph. I have chosen the axes to be momentum and velocity, with mass determining the gradient or slope of the line.
Now, if we want to calculate the area under the graph, this is simply the area of a triangle: half the base times the height. This holds true regardless of the gradient, so the area is ( frac{1}{2} times P times V ). We can substitute the original equation for ( P ) since ( P = mv ), which gives us the area as ( frac{1}{2} times M times V times V ) or ( frac{1}{2} mv^2 ). You may already be familiar with this equation; it represents the kinetic energy of a moving body. The mathematics we just did shows how these two equations are related, and in physics, we refer to this process as a derivation.
A derivation is essentially how you develop new physics by combining existing knowledge in a new way to describe something novel. Up to university level, derivations typically involve two types of mathematics: algebra and calculus. Algebra involves rearranging equations or combining them, like when we substituted ( P = mv ) in the previous example. Calculus consists of two parts: integration and differentiation. Integration is used to find areas under curves, while differentiation is used to find the gradients of curves. The purpose of calculus is to describe how certain quantities change in relation to others, which is a common occurrence in physics, making calculus an incredibly useful mathematical tool.
In my example, we didn’t actually use calculus because the graph is a straight line, allowing us to use simple geometry to find the area. Calculating the gradient of a straight line is straightforward; it’s just the height divided by the width of the triangle, which in this case is the mass ( M ). However, things become more complex when dealing with equations that have curved lines, like the plot of kinetic energy we derived, which is curved due to the ( V^2 ) term. In such cases, we need to use integration and differentiation.
I won’t have time to teach you all of calculus in this video, but I can give you a brief overview. The idea behind integration is to approximate the area under a plot using a series of rectangles. Each area can be expressed as ( frac{1}{2} MV^2 ), which gives the height times a small width, referred to as ( Delta V ). The total area is the sum of all these rectangles. The key to integration is that as you reduce the width of these rectangles, the approximated area becomes more precise and reaches perfection when the width approaches zero. We then rewrite this sum as an integral.
When we perform this integration, we arrive at a new equation. Interestingly, as physicists, we don’t have a specific name for the quantity we’ve just found, so we can name it whatever we like. This is often the case in physics when new concepts are discovered; physicists must find a way to describe them, which is why the field has so many unique terms.
Differentiation is essentially the opposite of integration. Here, we approximate the gradient using tiny triangles at each point. We express the gradient as ( frac{Delta left(frac{1}{2} MV^2right)}{Delta V} ). Again, the approximation becomes perfect when these triangles are infinitely small, leading us to a differential equation. The solution to this equation is ( MV ), bringing us back to where we started. Thus, differentiation is an inverse process to integration.
This gives you a taste of calculus, but if you want to learn more, I recommend the playlist on calculus by 3Blue1Brown. It’s an excellent resource that goes into much greater detail. The entire channel is fantastic!
Calculus is often taught as an abstract set of mathematical rules, but for physicists, it is essential to our daily work. Just as a plumber uses a wrench, a physicist relies on calculus. When learning physics in school, you are often given equations to memorize and apply in various ways, but the origins of those equations are rarely explained. This is understandable due to time constraints in school, but understanding the derivations of these equations can be incredibly satisfying.
When I was in university and learned calculus, I found it rewarding to understand where these equations came from and how they relate to one another. Being able to apply calculus to derive my own equations felt like a significant achievement. If you ever have the chance to learn calculus, I highly recommend it, even just for personal interest. It is definitely applicable, and understanding how to derive new physics is a compelling reason to learn.
As mentioned, algebra and calculus are foundational mathematical tools for physicists, but they are not the only ones. Depending on your path in physics, you may need to learn additional mathematics, such as matrix algebra, vectors, and tensors.
If you’re interested in learning more about derivations in physics and applying them to new situations, I recommend checking out the sponsor of this video, Brilliant.org. It’s a website where you actively solve problems broken down into manageable sections, encouraging careful thought and helping you build intuition about how different parts relate to each other. This approach makes it easier to apply knowledge in novel situations, rather than just memorizing equations.
If that sounds interesting, check out Brilliant.org/do s. The link is also in the description below. As an added bonus, the first 200 people to sign up can receive a 20% discount on the annual premium membership, which unlocks all of their content. Thank you for watching this video! I apologize for the delay since the last one; I’ve been busy with various projects, including a new book. There will be at least one more video this month, as I have a lot planned.
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This version maintains the essence of the original content while removing any informal language and ensuring clarity.
Momentum – The quantity of motion of a moving body, measured as a product of its mass and velocity. – In physics, the momentum of an object is conserved in an isolated system, meaning that the total momentum before and after a collision remains constant.
Calculus – A branch of mathematics that studies continuous change, involving derivatives and integrals. – Calculus is essential for understanding the motion of objects and the forces acting upon them in physics.
Integration – The process of finding the integral of a function, which is the reverse process of differentiation. – In physics, integration is used to calculate quantities like the area under a curve or the total displacement of an object over time.
Differentiation – The process of finding the derivative of a function, which measures how a function changes as its input changes. – Differentiation is used in physics to determine the velocity of an object by differentiating its position with respect to time.
Area – The measure of the extent of a two-dimensional surface or shape in a plane. – In calculus, the area under a curve can be found using definite integrals, which is crucial for calculating work done by a force in physics.
Gradient – A vector quantity that represents the rate and direction of change in a scalar field. – The gradient of a potential energy field in physics gives the force acting on an object at any point in the field.
Derivation – The process of obtaining a result from known principles through logical reasoning or mathematical operations. – The derivation of the equations of motion is fundamental in understanding the dynamics of systems in classical mechanics.
Velocity – The speed of something in a given direction. – In physics, velocity is a vector quantity that describes the rate of change of an object’s position with respect to time.
Mass – A measure of the amount of matter in an object, typically in kilograms or grams. – In physics, mass is a fundamental property of an object that determines its resistance to acceleration when a force is applied.
Kinetic – Relating to or resulting from motion. – The kinetic energy of an object is directly proportional to its mass and the square of its velocity, illustrating the energy associated with its motion.
