Lecture notes · Physics: Mechanics · Lecture 1

Course Introduction and Newtonian Mechanics

Yale PHYS 200 Fundamentals of Physics I, lecture 1, explained in plain English

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Part 1 · 0:00–10:00

Professor Shankar introduces the year-long physics course, its topics, how it is run, how it is graded, and how to do well in it.

Key ideas

The course covers the big ideas of physics, from Galileo and Newton to the 20th-century revolutions in relativity and quantum mechanics. 0:00
The topics were picked because physicists find them fascinating. Many of them can be useful in ways you cannot predict, even if you plan to be a doctor. 0:48
Class meets on Mondays and Wednesdays. Homework is handed out on Wednesday and is due before class the following Wednesday. 3:26
Grades are 20% homework, 30% midterm, and 50% final exam. Under the Amnesty Plan, the final exam alone can be used if it gives you a higher grade. 4:34
Homework is the most important way to learn physics, because working a problem yourself shows what you know and what you do not. Watching the professor solve one is not enough. 8:36

Words to know

relativity: Albert Einstein's theories about space, time, and motion, especially for objects moving very fast.
quantum mechanics: The branch of physics that describes how very small things like atoms and particles behave. Their behavior is different from everyday objects.
momentum: How much moving mass an object has. It depends on how heavy the object is and how fast it moves.
collaboration: A group of people working together on one project, where each person contributes a fair share.

Example

Think of learning to play a guitar. Watching a teacher play a song makes it look easy, but you only learn it when your own fingers try the chords. Homework in this course works the same way: it shows you where your understanding is weak.

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Part 2 · 10:00–20:00

The professor explains how the course works: where to get help, how to behave in class, and what math you need to know.

Key ideas

Understanding the material matters more than just getting the answers to problem sets. You should be able to explain how each problem is solved to others. 10:00
A TA (teaching assistant) is in the undergraduate lounge every day. The lounge has sofas, computers, and printers, and it is a good place to work on problems with other students. 10:16
Please don't talk to your neighbors during lecture. The professor reads students' faces to see if they follow him, so side talk distracts him. 11:04
Stop the professor any time you don't follow something. Not following does not mean you are weak at the subject, and your question may help the whole class. 15:25
The course is calculus-based. You need to know basic derivatives and integrals, and you must know trigonometry, such as sine and cosine, without looking it up. You do not need to memorize every trig identity. 18:17

Words to know

derivative: A number that shows how fast something is changing at a given moment.
integral: A way of adding up many small pieces to get a total, which is the opposite of a derivative.
trigonometric identities: Equations about angles and sine, cosine, and similar functions that are always true.
syllabus: The plan for a course that lists the topics and the order they will be covered in.

Example

Think of a cooking class where the teacher asks you to explain each step as you go. If you only copy the final dish without knowing why each step works, you will struggle when you cook alone. This course asks for the same kind of understanding.

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Part 3 · 20:00–30:00

Newtonian mechanics uses what you know right now about an object to predict where it will be later, and it splits into describing the present (kinematics) and explaining the changes (dynamics).

Key ideas

Physics aims to predict the future from the present. Scientists first choose a part of the world to study, then ask what information they need at the starting time to make that prediction. 23:35
Only the facts that matter are needed. For a thrown object, that means where it starts and how fast it starts moving. Its color does not matter. 25:31
Kinematics is a complete description of the present. It lists where things are and how fast they are moving, without asking why. 26:32
Dynamics explains how things change and why. For a falling object, the reason is gravity pulling it down. 26:59
To keep things simple, the object is treated as a mathematical point with no size. It is moved along a line, like a bead on a wire. 28:28

Words to know

Kinematics: Describing how something is right now: where it is and how fast it is moving, without asking why.
Dynamics: Explaining why things move the way they do and why they change, such as a force like gravity pulling something down.
Mathematical point: An imaginary object that has no size and no shape, so you never have to think about turning it around.
Initial conditions: The facts about a thing at the starting moment, such as where it was released and how fast it was going.

Example

Imagine tossing a ball to a friend. If you know where the ball leaves your hand and how fast it leaves, you can predict roughly where it will land. Your friend does this too, which is why they can move their hands to catch it.

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Part 4 · 30:00–40:00

This part explains how to describe motion with a position-versus-time graph, how velocity and acceleration are found from it, and why constant acceleration matters.

Key ideas

Motion needs units. Length is measured in meters and time in seconds, and every answer must carry the right units so it has a clear meaning. 30:00
A graph of position x against time t shows where an object is at each moment. A rising line means movement to the right and a falling line means movement to the left. The graph does not show the object bobbing up and down. 31:21
Average velocity is the change in position divided by the change in time. It can hide detail, because an object that goes out and comes back has an average velocity of zero even though it moved. 32:52
Velocity at one instant is found by making the time gap smaller and smaller. On the graph, it is the slope of the line that just touches the curve at that point. Acceleration is the derivative of velocity, which is the second derivative of position. 35:32
This course focuses on constant acceleration. Things falling near Earth's surface have the same constant acceleration, g, which is 9.8 meters per second squared. Finding position from acceleration is integration, which is the reverse of differentiation. 38:54

Words to know

derivative: How fast something is changing at one exact moment. It is the slope of a graph at a single point.
average velocity: The total change in position divided by the time it took. It gives the overall picture but may miss the details of the trip.
acceleration: How quickly velocity changes. It is found by taking the derivative of velocity.
integration: Working backwards from a rate of change to find the total amount. Here it means guessing the position function whose second derivative is the given acceleration.

Example

A car's speedometer needle says 60 at one instant, which is its velocity right now. If the car drove 60 miles in one hour, that is its average velocity for the trip, and the two numbers can differ. The speedometer is like the slope of the graph at one point, while the trip average is like the slope between the start and the end.

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Part 5 · 40:00–50:00

This part explains how to work backward from acceleration to find the position and velocity of an object moving with constant acceleration, and why starting conditions are needed.

Key ideas

Integration is the opposite of differentiation. Taking a derivative is a fixed recipe, but integration has no fixed recipe, so you have to guess a function and check it. 40:00
If an object has a constant acceleration a, its position has the form x(t) = x0 + v0 t + a t²/2. Any extra term that disappears after enough derivatives can also be added without changing the result. 42:30
Knowing the acceleration alone is not enough to know where an object is. You also need its starting position x0 and its starting velocity v0. These two numbers describe that particular object's history. 44:07
The constant x0 is the position at time zero, and v0 is the velocity at time zero. Both are chosen to match the situation you are studying. 45:57
You can also find the time from the velocities. Since v = v0 + at, the time is (v − v0) divided by a, so the speed of the object acts like a clock. 49:01

Words to know

integration: Working backward from a rate of change to find the original function. In this lesson it means guessing a function whose derivative matches what you know.
derivative: How fast a quantity is changing. You find it by comparing how much the function changes to how much the time changes.
acceleration: How quickly an object's speed changes over time. Here it stays the same at every moment.
initial velocity: The speed and direction an object has at the start, when time is zero. It is written as v0.

Example

Imagine throwing a ball up and catching it. You know where it started and how fast you threw it, and with those two facts plus gravity you can work out where it will be at any later moment. The same starting facts are what the lesson says you need to predict an object's path.

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Part 6 · 50:00–1:00:00

This part shows how to predict an object's height and speed using simple formulas, then uses a thrown ball to find its highest point and warns when those formulas stop working.

Key ideas

You can combine the formulas to remove time, which gives a formula linking speed, acceleration, and position: v squared equals v0 squared plus 2a times (y minus y0). 50:25
For an object with constant acceleration, its position is y0 plus v0 times t plus a half times a times t squared. The t squared term comes from the acceleration. 51:33
The thrown ball's full position formula is 15 + 10t - 5t squared. You can find its speed by taking a derivative of this formula. 54:05
A formula is only valid under the conditions it was built for. Once the ball hits the ground, the constant acceleration assumption fails, so the formula no longer applies. 55:00
At the highest point, the speed is zero. Setting the speed to zero gives a time of 1 second, and the height at that time is 20 meters. 57:16

Words to know

derivative: A way to measure how fast something is changing. For example, the derivative of position tells you the speed.
acceleration: How fast the speed is changing. A ball falling under gravity has a steady acceleration of about 9.8 meters per second every second.
velocity: How fast something is moving and in which direction. A negative value means it is moving downward.
quadratic: A formula where the highest power of the time is squared, like t squared. Its graph makes a curve.

Example

Think of a ball rolled up a hill. It slows down, stops for a moment at the top, then rolls back down. At the very top, its speed is zero, which is the same idea the lecture uses to find the highest point of the thrown object.

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Part 7 · 1:00:00–1:10:00

This part shows how the math of a falling object can give extra answers, and why those extra answers still matter.

Key ideas

You can pick any starting point, called an origin, to measure height from. The numbers change depending on the choice, but the physics stays the same. 1:01:05
To find when a falling object hits the ground, you set its height to zero. This gives a quadratic equation, which can have two answers. One answer is a negative time, which is before the clock started. 1:01:52
The math does not know about the building. It only knows the starting height, starting speed, and gravity, so it describes the path both forward and backward in time. 1:02:43
Paul Dirac found a second answer with negative energy in his equations. He could not throw it away, and it later pointed to anti-particles. 1:04:34
You can find the highest point without using time. At the top, the speed is zero, so you can use a formula that connects speed and height. This gives a maximum height of 20 meters. 1:08:04

Words to know

origin: The starting point you choose to measure everything from, like zero on a ruler.
quadratic equation: An equation with a variable squared, like t squared. It can have two answers.
momentum: How much movement a thing has, found by multiplying its mass by its speed.
derivative: A way to measure how fast something changes, such as how fast speed changes over time.

Example

Imagine a camera that filmed a ball rolling down a hill, starting from the middle. A physics formula could also tell you where the ball was before the camera started, even though no one filmed that part. The formula describes the whole path, not just the part you watched.

3 checked practice questions for this part are in the app.

Part 8 · 1:10:00–1:13:32

This part shows how the lecturer turns tiny changes in speed, distance, and time into the formula v² = v₀² + 2a(x − x₀), and when that formula is exactly true.

Key ideas

You can cancel the d in a derivative like dv/dt and treat the pieces as small changes, such as Δv and Δt. The lecturer says you should not cancel the d in a derivative itself, but this particular cancellation is legitimate. 1:10:00
You can multiply both sides of an equation by a small change in time, Δt. But that change has to be made vanishingly small in the end, because the formula is really the limit where the changes approach 0. 1:10:19
After multiplying by Δt, the left side becomes the change in v²/2 and the right side becomes a times the change in x. Adding up all the changes, which is what an integral does, gives v² = v₀² + 2a(x − x₀). 1:10:50
The equation is only exactly true when the changes are infinitesimal, meaning extremely small. If the time is a finite amount, like 1 second, the result will not be exactly true. 1:12:02

Words to know

derivative: A way of measuring how fast one quantity changes compared to another, such as how fast speed changes over time.
integral: Adding up many tiny changes to get a total. The lecturer says it means the same thing as adding up all the changes.
limit: The value something gets closer and closer to as a small number shrinks toward zero. The formula is really the limit where the changes go to 0.
infinitesimal: Extremely small, so small that it is almost zero but not quite. The lecturer uses it to describe the changes in the starting equation.

Example

Think of walking a long road by taking many tiny steps and adding them up to find the total distance. Each step is tiny, so the total is very accurate, but if you take one big step and call it the whole walk, the answer is only a rough guess.

3 checked practice questions for this part are in the app.

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. Open Yale Courses · CC BY-NC-SA 3.0 · not affiliated with or endorsed by Yale. These notes are shared under the same license.

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