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โš™๏ธย ap physics c: mech

ย ย >ย ย 

๐Ÿš—ย Unit 1

ย ย โ€ขย ย โฑ๏ธ5 min read

1.1 Kinematics Overview and Motion in One Dimension๐Ÿšด

Gerardo Rafael Bote

gerardo rafael bote

Caroline Koffke

caroline koffke

(editor)

โฑ๏ธ November 5, 2020

๐Ÿ“…

Overview

Everybody can see motion! However, motion is more than just moving. Motion is made up of a few different parts. For example, how can you describe the position of your body in relation to time? How can you tell that an object is faster than another? These are some of the questions physicists ask when studying kinematics. This is a MAJOR part of the course as the rest of AP Physics C: Mechanics has some sort of foundation within this unit.

Big Idea

  • Change:ย Interactions produce changes in motion.
    • When rolling down a hill, why do you go faster the farther you go?
    • If you want to throw a stone as far as you can, why should you throw the stone higher?

Exam Impact

Unit 1 will cover approximately 14%-20% of the exam and should take around 22, 45-minute class periods to cover. The AP Classroom personal progress check has 15 multiple choice questions and 1 free response question for you to practice on.

1.1: Kinematics: Motion in One Dimension๐Ÿšด

To explain motion, you need to know the following terms:

  • The position of an object is its location within a certain time frame.
  • The displacement of an object is its relative change in position in some reference to some origin. (ฮ”x=xโˆ’x0)
  • The distance of an object is the length of the path covered by the object.
  • The velocity of an object is the rate at which the object's displacement changes over time.
  • The speed of an object is the rate at which the object's distance changes over time.
  • The acceleration of an object is the rate at which the object's velocity changes over time.

โš ๏ธBut, wait... aren't displacement and distance the same thing? Aren't speed and velocity the same thing, too? Well, not exactly. You need to consider that displacement and velocity (and acceleration) are both vector quantities, meaning that they bring both a numerical value (i.e., magnitude) and a direction. Scalar quantities, like distance and speed, only bring a magnitude (no direction included) and are always positive.

For example, in the picture below, the bicyclist's distance traveled would be measured in path A (the road); on the other hand, the bicyclist's displacement from his/her house to the factory would be measured in path B.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-nAF4grirCugk.png?alt=media&token=fc579ed7-c4b0-4ec1-89af-b2b4f9f558ce

Image from Dan Levine

For vector quantities, a positive number (+) indicates that the object's rate or direction is up or right. A negative number (-) indicates that the object's rate or direction is down or left.

For those who want to visualize displacement, velocity, acceleration as vector quantities, refer to the graphic below:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-PPVKD3n7ydYj.png?alt=media&token=0022a620-cb08-42a7-86ce-411a602502ce

Image from Online Math Learning

Average velocity is a comparison between the displacement and the time required for the displacement to occur. It is approximating motion since the velocity could have increased or decreased during the specified time interval.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-09-11%20at%209.48-y2oq4z9zf3NU.png?alt=media&token=89840fc0-1f8a-4493-a5ee-a04f5112b2f7

Instantaneous velocity, on the other hand, is the velocity that was measure at a certain time frame. Deriving a position function will allow one to detect an object's velocity at some time frame without using average velocity to estimate it.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-09-11%20at%209.48-eDyBJAF4FRSY.png?alt=media&token=194ebd0d-73c9-499d-bd3c-407508001608

There are many ways to represent the relationship between velocity (v), acceleration (a), position (x)๏ปฟ, and time (t)๏ปฟ. Know the following equations to understand how they relate

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-09-11%20at%209.42-A5rWkcfWAdPY.png?alt=media&token=fab472e4-5d8f-4761-aa01-6a90e3a10824

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-kZ0z51X0Y6VP.png?alt=media&token=6ca6ce74-b106-4fdd-b8cd-ebebbb3c4bf3
TIP: If you need to solve for a quantity from the above equations, make sure that you can solve it given that you have the other variables filled in! For example, in the 2nd equation, if you have the final velocity, the time interval, and the acceleration rate, then you can solve to figure out the initial velocity!

There is also one more relationship that you should consider:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-09-11%20at%209.42-U0SfaVigoVjY.png?alt=media&token=7faa4ae3-5e77-4851-8e85-bccb716c5d16

What the equation shows above is that the function representing an object's acceleration is the same function that represents the 1st derivative of the velocity function and the 2nd derivative of the position function. It works the other way too:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-09-11%20at%209.57-Yp303rpGjYWk.png?alt=media&token=96c3a60c-5428-4f4f-bd1c-c7f484418757

The position function is represented by the integral of the velocity function and the double integral of the acceleration function.

โš ๏ธBut, wait... I don't know how to differentiate and integrate... What? You don't know how? There is a reason why this course is called AP Physics C! C means Calculus! Don't worry, here are the basic rules for doing integrals and derivatives for this course:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-BpKmMYQi5xRD.png?alt=media&token=fa07711c-0996-4f03-9c08-4c6c065adbe7

If you're given a kinematic graph (i.e., a graph containing time and some other measure), use the following table to figure out what part of the graph you need to solve for:

Kinematic Graphs

Types of Kinematic GraphsArea Under CurveSlopeMagnitude of Y-value
Position VS. TimeN/AVelocityDistance from detector or starting position
Velocity VS. TimeChange in PositionAccelerationSpeed of Object
Acceleration VS. TimeChange in VelocityJerk* (Not Tested in AP Exam)Acceleration

Use the following table to describe the vector quantities:

Vector Quantities

Vector Negative (-)Zero (0)Positive (+)
DisplacementYou are now south, west, left, or in the -x or -y direction of your starting position.You are back at your starting position.You are now north, east, right, or in the +x or +y direction of your starting position.
VelocityYou are traveling south, west, left, or in the -x or -y direction.You are at rest.You are traveling north, east, right, or in the +x or +y direction.
AccelerationIf your velocity > 0, your speed is decreasing in a positive direction. If your velocity < 0, your speed is increasing in a negative direction.You are at restย ORย you are moving at a constant velocity.If your velocity > 0, your speed is increasing in a positive direction. If your velocity < 0, your speed is decreasing in a negative direction.

Practice Question:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rFCVvEHocXW3.png?alt=media&token=b2b17c86-f658-4f63-8a6d-2acc3ce414d5

Image from collegeboard.org

Answer

The correct answer is D. To show a constant acceleration on a velocity vs. time graph, you need to consider the slope of each line segment. For constant acceleration, it is shown through a straight slope (not curved) on a velocity vs. time graph; therefore, QR cannot be it. For an acceleration to be nonzero as well, the slope of the velocity vs. time graph cannot be horizontal, marking out PQ and TU. Therefore, RS & ST satisfy the question.

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