Chapter 2 Part 1 Displacement, Vectors, Speed, Velocity, Acceleration

Prof. Clements Notes/Hints for Physics Courses

Chapter 2 Open Stax College Physics or Most Introductory Physics Courses

Part 1: Displacement, Vectors, Speed, Velocity, Acceleration

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know after this unit: displacement, delta, scalar, vector, speed, velocity, acceleration, average, instantaneous

The study of motion will occupy a significant portion of the first semester of a physics class. This chapter presents the description of motion. Later chapters will present the cause of motion (force). It is important to have a good foundation on the description of motion so you can better understand situations where the force needs to be analyzed. A simplification in this chapter is that we will only let objects move in one line. The object may move back and forth, or up and down, but the motion will be in a straight line.

Displacement

In order to describe the motion of an object we need to make measurements of the position as a function of time. Position measurements are made in a reference frame. A straight line with a zero mark at some location can be a reference frame.

Displacement is the change in position of an object. When working in one dimension the direction of change is indicated by a + or - in front of the size of the displacement.  X = X_f - X_o where X_f is the final position and X_o is the initial position. The  (delta) symbol tells us to subtract two numbers. You only need the final and initial position values. We do not need to keep track of the intermediate positions of the object.

A certain person walks 80 cm North, 60 cm South and 30 cm North. What is the displacement?

Answer:size of the displacement: 50 cm direction of the displacement North

Note that displacement answer is incomplete if it is missing either the size (magnitude) of the displacement or the direction.

Along our X axis we will use + to indicate displacement in the positive direction. A - sign will indicate displacement in the negative direction.

In the real world objects seldom travel on a perfectly straight line with no reversals of motion. As real objects move they will move along curves and straight lines and may back up. The total path length covered is called the distance traveled.

How does distance differ from displacement?

Distance traveled is usually larger than the magnitude of the displacement. Remember, displacement is measured in a straight line.

Distance traveled can be equal to the magnitude of the displacement if the motion is in a straight line with no reversal of motion.

Vectors, Scalars, Coordinate Systems

Quantities that have both a size (magnitude) and direction are called vectors. Displacement is a vector. The + or - sign is enough to indicate direction. You don't have to use words such as left or right to describe the direction of the displacement.

Examples of vectors: displacement, velocity, acceleration, force, electric field …

Examples of scalars: temperature, mass, speed ...

The main difference between a scalar and a vector is that the scalar does not have a direction.

You will be using the Cartesian coordinate system (three axes at right angles to each other, X Y Z). Usually discussions are restricted to motion in at most 2 dimensions. For horizontal motion I will describe positions along a X axis. For vertical motion I will describe positions along a Y axis.

As you work on problems you must choose the positive direction and the negative direction. You should label this direction with a + on the sketch you make for the problem. Once you select the + direction you must make sure all of the values you write down from the word problem are consistent with the + direction you have selected. i.e. If you choose the + direction to the right and the object moves to the right 5 meters in 2 seconds you would use a displacement value of + 5 meters.

You will often have some freedom in where to place the origin of the coordinate system. The placement of the origin does not affect the results of motion calculations. i.e. For an object moving on a straight track it doesn't matter if we choose the origin (X = 0) of the coordinate system to be at the left edge of a track or at the middle of the track or at the right edge of the track. The position numbers do change, but the important quantities of velocity and acceleration will be the same regardless of the location of the origin of the coordinate system.

Time, Velocity, Speed

When time has elapsed there will be a change. e.g. the hands of clock have moved, your heart has made a beat, a leaf has moved, a sound has been detected, the Earth has moved in its orbit around the Sun, etc. The time for motion to take place will be used in calculating velocity and speed. We will often refer to some clock and say the time on the clock is 0 seconds at the start of the motion and equal to some value, t, at the end of the motion. The time interval, Δt = t_f -t_o, will simplify to just t_f when

t₀ has a value of 0.

The average velocity is found by dividing displacement by time. Because displacement is a vector the velocity is also a vector. If an object has a displacement of +8 meters in 4 seconds the average velocity is +2 meters/second. The “+” sign indicates the direction is to the right on the axis.

Instantaneous velocity, v, is the velocity of the object at a particular instant of time.

Consider a car stopped a red light on a city street. Suppose that after the light turns green the driver presses lightly on the gas pedal for two seconds and the car starts moving faster and faster until it reaches some final velocity. During this two second time interval after the light turns green the car will have some average velocity (that will be smaller than the final velocity). The instantaneous velocity will be different at each instant of time until the car reaches its final velocity. The speedometer of the car (approximately) gives the velocity at each instant of time.

Instantaneous velocity, v, can be found by calculating  X /  t as  t approaches 0. This requires the use of Calculus.

Another term that describes motion is speed. Speed and velocity are usually much different quantities. The distinction in these quantities comes from using the distance traveled or the displacement.

The average speed = (total distance traveled) / (time required)

Speed is a scalar because it does not have a direction.

One can also calculate the instantaneous speed by considering a very small time interval.

How far did you drive to come to your school to class?

How much time was required for the trip?

Calculate your average speed distance/time

Imagine a map of your journey and estimate the displacement. Calculate your average velocity.

displacement/time → ______________ ____________

(The first blank is for the magnitude of the average velocity and the second blank is for the direction.)

The average velocity value usually be smaller than the average speed. This occurs because the straight line displacement magnitude is usually smaller than the distance traveled (that includes curves, etc.)

The size of the instantaneous velocity is called the speed of the object. Speed does not have a direction. Quantities that do not have an associated direction are called scalars. Speed is a scalar.

Acceleration

The measure of the rate of change of velocity is called acceleration. The average acceleration is found by dividing the quantities v and t.

a_avg = v/ t The direction of the acceleration should be specified, but this can be done with + and - .

Suppose an object has a velocity of + 2 m/s at time = 0 seconds and then has a velocity of +9 m/s at a time of 2 seconds. What is the average acceleration?

Answer: The average acceleration is ( 9 m/s - 2 m/s) / 2 seconds or (7 m/s) / 2s or +3.5 m/s²

The meaning of an acceleration of +3.5 m/s² is that the velocity is increasing by 3.5 m/s every second. If the acceleration value is constant, the velocity would be 2 m/s at time 0, 5.5 m/s at time = 1 second, and 9 m/s at time = 2 seconds. If the acceleration continues the velocity would be 12.5 m/s at time = 3 seconds.

Some texts discuss deceleration. The key concept is that if the acceleration has a - (negative) value then the velocity is becoming more negative each second. Acceleration values will be + and - in homework and exam problems. e.g. At time = 0 the velocity is 15 m/s and the acceleration is -4 m/s². At time = 1 second the velocity would be 11 m/s. At time = 2 seconds the velocity would be 7 m/s, etc.

Note that the mathematical procedure used to calculate average velocity is the same as the procedure used for calculating average acceleration: a difference is divided by a time interval.

The instantaneous acceleration is the acceleration value at an instant in time.

An estimate can be made for the instantaneous velocity at some desired time, t, from the graph of position vs time. The method involves: 1) Draw a straight line that grazes the graph of position vs time at the desired time, t. This line is often called a “tangent” line. On each side of the place where the straight line touches the graph of position vs time, there will be equal gaps between the straight line and the graph of position vs time. See your textbook for examples. 2) Calculate the slope of the straight line (rise/run). This is the instantaneous velocity.

A similar procedure for the graph of velocity vs time will yield values for the instantaneous acceleration.

These calculations for instantaneous velocity and acceleration will be approximate as you cannot draw a “perfect” tangent line, nor calculate the exact value of the slope. You will use equations later in this chapter to obtain better values for the velocity and acceleration.

Your textbook probably shows some graphical representation of motion with graphs of position vs time, velocity vs time, and acceleration vs time. You should work through some of the examples in your textbook and ask your instructor if you have questions.

You should be able to make approximately accurate graphs of velocity and acceleration if you are given the graph of position vs. time. Some instructors do not have you make numeric calculations in creating the velocity or acceleration graphs. Important concepts regarding motion can be obtained by just estimating the value of the velocity and acceleration (especially if it is +, 0, or -) at various times. The graph is then quickly sketched.

Check your understanding of the concepts with these statements.

1. If the velocity is zero the acceleration does not have to be zero. e.g. If you toss a ball upward, at the instant the ball is at its highest position the velocity is zero but the acceleration is still -9.8 m/s² . Think this through. If you say the acceleration is zero for the ball at the top of its motion, then the velocity would not change (remember acceleration = change in velocity/time), and the position of the ball would not change (if velocity is zero the position is not changing). The ball would just hang in the air some distance above the ground!

This point is very important. Understanding the next two statements is key to understanding motion.

Velocity does not control the value of acceleration!

Acceleration controls the future value of the velocity!

2. TRUE or FALSE If the velocity is positive then the acceleration is positive.

Answer: False. The acceleration could be -, 0, or +. You must subtract two velocities to obtain acceleration. One velocity number is insufficient data to claim a certain value for the acceleration.

3. TRUE or FALSE If the acceleration is zero then the velocity is zero.

Answer: False. The velocity could be some constant value such as 5 m/s. The acceleration is found from ( 5 m/ s - 5 m/ s ) / time interval, which is 0 m/s² .

What is the meaning of an acceleration of - 3 m/s² ?

Answer: The velocity is decreasing by 3 m/s every second.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.Reading Guide Chapter 2 Sections 1 to 4 Page 37 in pdf file OpenStax College Physics