Kinematics is the study of motion through describing and relating the position ($\overrightarrow{r}$), velocity $(\overrightarrow{v})$, and acceleration ($\overrightarrow{a}$) of a system. Average Quantities involves taking snapshots in time of an initial and final state of a system and finding the average of the above quantities. This section is also intended to practice vector mathematics. Operations, like adding or subtracting vectors, are a very important skill in almost all of physics, including kinematics.

The difference between speed and velocity.

https://www.youtube.com/watch?v=X4Wxd4m-QVc

## Pre-lecture Study Resources

Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.

### BoxSand Introduction

## Average Quantities | Position and Displacement

You can define a position vector by it's x and y components.

$\overrightarrow{r}=\langle x, y \rangle$

You can also define the change in position (or displacement) as the final position minus the initial position.

$\Delta \overrightarrow{r}=\overrightarrow{r}_f - \overrightarrow{r}_i$

The vector diagram for a situation like this is below. In the case of position vectors, it is also what a birds-eye view might see. The axis, with origin * o*, defines the point <0, 0> and the positive $\hat{x}$ and positive $\hat{y}$ directions. (Note: the symbol $\hat{x}$ is called

*x hat*and can be thought of as a way to show which direction is the positive x direction. It is technically a unit vector but that formality is unnecessary at this point.) You could imagine an object that begins at point

*and after some time, ends at point*

**q***. The change in position, labeled $\Delta \overrightarrow{r}_{qp}$ points from the initial position*

**p***, to the final position*

**q***.*

**p**

The first important feature of the above diagram is that the position vector $\overrightarrow{r}_q$ can be added to the change in position vector $\Delta \overrightarrow{r}_{qp}$ using the head to tail method. The vectors don't even have to be transposed because they are already setup in the correct orientation. The resultant of that addition would be the vector $\overrightarrow{r}_{p}$. The next lesson is that you can now do regular algebra on that expression and solve for any of the vectors. The change in position $\Delta \overrightarrow{r}_{qp}$ has been derived above. The expression could also have been algebraically manipulated to solve for $\overrightarrow{r}_q$ in terms of the other two. For more algebraic rules regarding vector equations, see BoxSand's pre-lecture videos.

The term Average here refers to the fact that we are only concerned with snapshots in time and the object didn't necessarily travel along the change in position vector direction. It could have followed a curved path because all we know is that it was initially in one location and is now in another. So this could be referred to as the average change in position, but we just call it the change in position.

## Average Quantities | Average Velocity and Acceleration

Average velocity is just a small step away from change in position.

$\overrightarrow{v}_{average}=\frac{\Delta\overrightarrow{r}}{\Delta t}$

Here $\Delta t$ is the change in time from the initial to final position. It's important to note that when a vector quantity, such as $\overrightarrow{v}_{average}$ is set equal to another vector quantity, such as $\Delta\overrightarrow{r}$, and the only other quantity in the relationship is a scalar, such as $\frac{1}{\Delta t}$, then both vector quantities must point in the same direction. So $\overrightarrow{v}_{average}$ points the same direction as $\Delta\overrightarrow{r}$. You can also apply the same algebraic rules to this equation and solve for any of the quantities.

## Key Equations and Infographics

To see how this looks when analyzing the average acceleration, see the pre-lecture videos below.

### BoxSand Videos

## Required Videos

## Suggested Supplemental Videos

### OpenStax Reading

### Fundamental examples

1. A ball is set on the end of a table which is 6 meters long. The ball rolls to the other end of the table in 1 minute. What is the average velocity of the ball, in meters per second?

2. A ball is thrown straight down off a cliff with an initial downward velocity of $5 \frac{m}{s}$. It falls for 2 seconds, when the downward velocity is recorded as $24.6 \frac{m}{s}$. What is the balls average acceleration($\frac{m}{s^2}$) during that time?

3. An object moving in a straight line experiences a constant acceleration of $10 \frac{m}{s^2}$ in the same direction for 3 seconds when the velocity is recorded to be $45 \frac{m}{s}$. What was the initial velocity of the object?

CLICK HERE for solutions

## Post-Lecture Study Resources

Use the supplemental resources below to support your post-lecture study.

### Practice Problems

BoxSand's Quantitative Practice Problems

BoxSand's Multiple Select Problems

Practice Problems: Multiple Select and Quantitative.

Recommended example practice problems

- Set 1: 8 Problem set with solutions following each question. Be sure to try to question before looking at the solution! Your test scores will thank you. Website
- Set 2: Problems 1 through 6. Website

Example Sets with solutions

- 8 Problem set with solutions following each question. Website
- 6 Problems with key. Website
- University of Wisconsin-Green Bay | Components of a Vector, Average Velocity.
- Vector addition practice problems with solutions: Web

Worked Examples

- Flipping Physics average velocity

https://www.youtube.com/watch?v=WpdYBnyZOm8 - Vectors

https://www.mathsisfun.com/algebra/vectors.html

## Additional Boxsand Study Resources

### Additional BoxSand Study Resources

### Learning Objectives

## Summary

The objective is to introduce vector mathematics in multiple representations through the use of physical quantities such as position, displacement, average velocity, and average acceleration.

## Atomistic Goals

Students will be able to...

- Recognize that a vector consists of a magnitude and direction, it is multiple pieces of information, while a scalar is just a number (one piece of information).
- Identify the difference between a vector and it's magnitude, specifically speed and velocity or distance and displacement.
- Draw a vector and identify in the drawing the magnitude, direction, and Cartesian components.
- Mathematically define vectors in terms of Cartesian (x, y) components or magnitude and direction and go between these representations.
- Recognize that a vector can be translated left, right, or up and down, so long as the magnitude and direction remain unchanged.
- Add (and subtract) vectors both pictorially with the head to tail method or parallelogram method and recognize that the resultant is equivalent in both.
- Add (and subtract) vectors using a vector operation diagram.
- Create vector equations from the graphical representation and vice versa.
- Add (and subtract) vectors algebraically in component form.
- Demonstrate that all of the vector rules are still applicable in one dimension.
- Demonstrate the correct mathematical use of vector algebra rules.
- Define the characteristics of position, displacement, average velocity, velocity, and average acceleration.
- Use vectors to determine distances as opposed to using geometry, e.g. finding the displacement between two positions and then the magnitude of that vector is the distance between them.
- Show that the position of an object depends on the origin but the change in position, the velocity, and the acceleration do not.
- Understand the directional relationship between two vectors when the vectors are related to each other by a scalar.
- Demonstrate that the direction of the displacement is the same as the direction of the average velocity.
- Demonstrate that the direction of the change in velocity is the same as the direction of the average acceleration.
- Identify kinematic terminology: Displacement is the same as the change in position.
- Identify kinematic terminology: Distance is the same as the magnitude of the displacement.
- Identify kinematic terminology: Speed is the same as the magnitude of velocity.
- Identify kinematic terminology: Time interval is the same as the change in time.
- Identify kinematic terminology: The magnitude of the acceleration is called the magnitude of acceleration.
- Show that the total distance traveled is not the same the magnitude of the displacement
- Go between the physical and mathematical representation.
- Calculate quantities involving the average velocity
- Calculate quantities involving the average acceleration
- Calculate quantities involving the position
- Calculate quantities involving the change in position
- Show when the instantaneous velocity is equal to the average velocity and that it points in the direction of motion.
- Show that the change in any vector points from initial to the final vector when they are placed tail to tail.
- Show that in 1 dimension, when an object is speeding up, the acceleration points in the same direction as the velocity but when it is slowing down they point in opposite directions.
- Extrapolate the rules for 2D vectors into a 3D application.

### YouTube Videos

This short video talks about average speed and velocity.

https://www.youtube.com/watch?v=ZL_pxasSd-o

This Khan Academy video does a great job of showing the difference between instantaneous speed, velocity, for one dimensional motion. There's a bit towards the end that involves a bit of history and a note about calculus, the video does not use that calculus though, when it does an example.

https://www.youtube.com/watch?v=pfTTHx9kCHk

Doc Schuster has created a wonderful series on youtube to help students understand their physics. This video is a fairly in-depth example involving average velocity. This video also shows how you are not anchored to using the average velocity equation as it is given, you are free to preform any algebraic manipulation you would with any other equation.

https://www.youtube.com/watch?v=CaH57mVYhxU

Video and example problem for speed versus velocity.

### Simulations

This interactive quiz from The Physics Classroom asks you to simply name that motion. This simulation should solidify difference between average velocity and instantaneous velocity as well as the effect of acceleration on a moving object. *Note, the frame of the animation is a bit small when you first run it, you can click on the lower right corner and drag to make the animation frame larger.

For additional simulations on this subject, visit the simulations repository.

### Demos

https://www.youtube.com/watch?v=WpdYBnyZOm8

For additional demos involving this subject, visit the demo repository

### History

Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Physics Fun

Oh no, we haven't been able to post any fun stuff for this topic yet. If you have any fun physics videos or webpages for this topic, send them to the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Other Resources

This page from The Physics Classroom will help in determining the difference between average and instantaneous quantities, representing vector quantities, and calculated average quantities.

Another page from The Physics Classroom that helps with the same as the above, but this time focusing on Acceleration.

Hyper Physics is also another resource that will be on almost every page. Each page on a subject is made to be a quick little note with the relevant information, making Hyper Physics a great reference page. This page for instance gives a concise reference for velocity, and average velocity.

## Resource Repository

This link will take you to the repository of other content on this topic.

## Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

### Assumptions

- Finding the change in position, average velocity, or acceleration only involves consideration of the initial and final state of the system during that time period.
- Knowledge of what caused a system to change its acceleration, namely the net force, will not be required in this section.
- A coordinate system should be defined in every case but if one is not explicitly written, it will be assumed positive $\hat{x}$ is to the right and positive $\hat{y}$ is upward.

### Checklist

1. Read and re-read the whole problem carefully.

2. Visualize the scenario. Mentally try to understand what the object is doing.

a. Motion diagrams are a great tool here for visual cues as to what the motion of an object looks like.

3. Draw a physical representation of the scenario; include initial and final velocity vectors, acceleration vectors, position vectors, and displacement vectors.

4. Define a coordinate system; place the origin on the physical representation where you want the zero location of the x and y components of position.

5. Identify and write down the knowns and unknowns.

6. Write out the mathematical representation for the physical representation you drew in step (3).

7. Carry out the algebraic process of solving the equation(s).

a. If simple, desired unknown can be directly solved for.

b. May have to solve for intermediate unknown to solve for desired known.

c. May have to solve multiple equations and multiple unknowns.

d. May have to refer to the geometry to create another equation.

8. Evaluate your answer, make sure units are correct and the results are within reason.

### Misconceptions & Mistakes

- Remember that speed is a scalar quantity which lacks direction, and velocity is a vector quantity which includes direction
- When given multiple velocities, lets say 3 velocities, and attempting t find the average velocity, you
**cannot**just add the velocities together and divide by three. You can find the average this way**only**if each velocity is traveled for the same amount of time, which will rarely be the case.

### Pro Tips

- Always ask yourself if the questions is asking for a vector or a scalar quantity.
- Always ask yourself it the information given is a vector or scalar quantity.
- Draw a physical representation for the situation. Often this includes vector operation diagram.

### Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

### Physical

### Mathematical

### Graphical

Below is a graph with position on the vertical axis and time on the horizontal. The curve shows a tangent line, or slope, representing in instantaneous velocity at one point. There is also a representation of the average slope and thus velocity between two points.

### Descriptive

### Experimental

We could go out to an ordinary city block and time how long it takes to walk from one corner of the block to the exact opposite corner. In this way, we could walk around the outside of the block and measure the distance traveled on foot around the block. Each side of the block would represent a position vector. From the mathematical representation section we would certainly know what to do with the rest information. Therefore, we can easily calculate the the average velocity.