1. Elementary (but not necessarily trivial) Concepts

Placement & displacement; position, distance & position; time, time intervals & clock readings; speed, velocity, average velocity & instantaneous velocity; acceleration, average acceleration & instantaneous acceleration: The Greeks, for all their mathematical skills and accomplishments in logic, failed to develop these concepts and hence failed to discover the laws of motion. If you have difficulties with these concepts, you are in good company, they eluded the best minds in the world for 2000 years.

The idea of model building: (The "physics" of physics)

When studying the motion of an object, what all must be considered ? For instance, if all we want to know is how many meters an object travels in a straight line in a set amount of time, do we need to know its shape, or whether it is rotating ? No. We will first study the motion of the whole object, and ignore its shape, or whether or not it rotates as it moves, so we will treat the whole object as if it were a single dot, or particle.

Mental Demo/Discussion:

Place a glider on an air track (friction free surface) and give it a gentle shove. Draw a picture of "strobe light photography" with equal time intervals between flashes. Note that we pick a spot on the craft and use it as the idealized point location for the whole craft.

Next incline the track to provide constant force, release the glider at the top and produce a motion diagram, then give it an initial push up the track from the bottom to provide motion in one direction with constant force in the other. or use fan cart

Discuss the location of the glider at different instants in time. Think about these different positions and develop the idea of a frame of reference, and position relative to the reference frame.

Position - How big is a location? (How long is a point?) Hint: How many points on an interval 1 unit long?

Position number - Distance from some stated reference point to a given location.

Displacement - Total description of an object's location relative to a known frame of reference. Distance from some stated reference point to a given location and the direction one must travel relative to a given base line to get there.

Notice that one could describe the position number of P1, P2, P3, P4, P5 relative to 01 simply by assigning a positive or negative number to the point. Positive numbers tells us that the point is to the right of the origin, and negative ones tell us that the point is the designated number of unit lengths to the left of 01.

How could we describe displacement of P1, P2, P3, P4, P5 from 01? Using the same numbers! Position and displacement relative to a designated point of origin are really the same thing.

However, to describe the position of P1, P2, P3, P4, P5 relative to 02 simply by using their distances from 02 would not really uniquely describe those locations, since there is literally an infinite number of points the same distance from 02 as each of these points.

How could we describe the displacement of P1, P2, P3, P4, P5 from 02?

To determine the position of points or displacement between points relative to an origin that is non-collinear with them requires two or more numbers, or, vector quantities.

Adding up consecutive displacements to obtain total displacement.

Suppose a person walks from P1 to P2 , then to P3. What is the total displacement (from P1)?

Play around with this a bit, walking from point to point to develop the idea of vector addition. Be sure to discuss correct labeling of vectors.

 

Next look at first displacement and final location and guess what second displacement is. Try to draw a graph of this difference.

Develop vector subtraction, and the idea that change in displacement Dr = r₂ - r₁ results from placing the two vectors r₁ and r₂ tail to tail. The change, or difference is the line segment connecting their heads. To get the change in displacement resulting from moving from point r₁ to the point r₂, place the arrow head on the line at r₂, to get the change in displacement resulting from moving from r₂ to the point r₁, place the arrow head on the line at r₁.

Notice that while displacement (location) depends upon the reference frame, the change in displacement is independent of the reference frame.

Time - Clock Readings - instants in time.
Time Intervals - the difference between two clock readings.

Note: All points on earth are fixed relative to each other in time.)

How long does the glider remain at any given location during its motion?
How long does a ball tossed up remain at any position of its flight?
How long does a ball tossed up remain motionless at the top of its flight?
How long does an pendulum spend at the end of its swing?
How long does any continuously moving object occupy a given location?
So! How long is an instant? Hint: How many "instants" in a second?

Definition: Event - clock reading together with a position.

A brief discussion of the meaning of the words "proportional to" and "uniform".
"is proportional to" can be said in several ways,
"is directly proportional to"
"varies as"
"varies directly as"
which means that the ratio of two particular quantities is always a constant.

Uniform - Basically means "the same everywhere", in physics, it means "something" stays constant in time.

- Used in the phrase "uniform circular motion", it can mean "the same change in position in each succeeding second", which means "something stays constant". (Discuss what.)

- Used in the phrase "Kinematic equations of rectilinear motion with uniform acceleration", it means acceleration is constant in time (the same each second).

- "Something" can seem a bit vague at first. Discuss the above experiments, decide if they are examples of uniform motion, and decide what "something" is constant.

Note: The following comes from Alan Van Huevelen 's Alps Kit

When first applying kinematic (motion) principles there is a tendency to use the wrong kinematic quantityÐto inappropriately interchange quantities such as position, velocity, and acceleration. Constructing a motion diagram should reduce this confusion and should provide a better intuitive understanding of the meaning of these quantities.
A motion diagram represents the position, velocity, and acceleration of an object at several different times. The times are usually separated by equal time intervals. At each position, the object's velocity and acceleration are represented by arrows. If the acceleration is constant throughout the motion, one arrow can represent the acceleration at all positions shown on the diagram. The motion diagrams for three common types of linear motion are described below.

Constant Velocity: The first motion diagram, shown in fig. 1.3, is for an object moving at a constant speed toward the right. The motion diagram might represent the changing position of a car moving at constant speed along a straight highway. Each dot indicates the position of the object at a different time. The times are separated by equal time intervals. Because the object moves at a constant speed, the displacements from one dot to the next are of equal length. The velocity of the object at each position is represented by an arrow with the symbol v under it. The velocity arrows are of equal length (the velocity is constant). The acceleration is constant because the velocity does not change .

Constant Acceleration in the Direction of Motion: The motion diagram in F1g. 1.4 represents an object that undergoes constant acceleration toward the right in the same direction as the initial velocity. This occurs when your car accelerates to pass another car or when a race car accelerates (speeds up) while traveling along the track. Once again the dots represent schematically the positions of the object at times separated by equal time intervals Dt . Because the object accelerates toward the right, its velocity arrows increase in length toward the right as time passes. The product a Dt) = Dv represents the increase in length (the increase in speed) of the velocity arrow in each time interval Dt. The displacement between adjacent positions increases as the object moves right because the object moves faster as it travels right.

Constant Acceleration Opposite the Direction of Motion: The motion diagram in Fig. 1.5 represents an object that undergoes constant acceleration opposite the direction of the initial velocity (this is sometimes called deceleration_-a slowing of the motion). For this case the acceleration arrow points left, opposite the direction of motion. This type of motion occurs when a car skids to a stop. The dots represent schematically the positions of the object at equal time intervals. Because the acceleration points left opposite the motion, the object's velocity arrows decrease by the same amount from one position to the next. We are now subtracting D v = a (Dt ) from the velocity during each time interval Dt . Because the object moves slower as it travels right, the displacement between adjacent positions decreases as the object moves right.

You should become so familiar with these motion diagrams that you can read a linear-motion problem and draw a reasonable diagram that represents the motion described in the problem. When you complete the mathematical solution to a kinematic problem later in the semester, you can see if your answer is consistent with the motion diagram.

Official Definition of Average Velocity:

Given two position numbers r₁ and r₂, relative to some defined frame of reference, and two clock readings t₁ and t₂, the average velocity of the object between r₁ and r₂ is Dr/Dt = (r₂ - r₁)/(t₂ - t₁).

What does it tell you ?

It tells how far and in which direction the object travels each second, on average, as it moves from r₁ to r₂.

Notice that since the time interval is the same between any two successive points in a motion diagram, a vector drawn between successive points could represent either change in displacement Dr between them or average velocity v over that time interval.

What does it NOT tell you?

Notice that average velocity is change in displacement / change in time, not total distance traveled /time (That's Average Speed).
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Analyze the following motion diagrams:
Exercise 1. Below is the motion diagrams for two different objects A and B. Clock readings are indicated in the boxes.

a) Do both have uniform velocity?
b) Which is travelling fastest?
c) Which travels for the greatest length of time?
d) Which travels the greatest distance ?
e) At which clock readings(instants) do they occupy the same location at the same time?

Exercise 2: Below is the motion diagrams for two different objects C and D. Clock readings are indicated in the boxes.

a) Do both have uniform velocity?
b) Which is travelling fastest to begin with?
c) Which travels for the greatest length of time?
d) Which travels the greatest distance ?
e) At which clock readings(instants) do they occupy the same location at the same time?
f) Is there any time interval over which they have the same average velocity?
g) Are they ever travelling the same speed at any clock reading?
h) Which would accurately indicate the speed of the object at clock reading 2, an arrow drawn from 2 to 3 for C or an arrow drawn from 2 to 3 for D?

This is actually a new concept, the speed of the object at a clock reading (instant), appropriately named, Instantaneous Speed. In exercise 4 below we will develop a systematic method for obtaining it. But first:

Exercise 3. Be careful with the word "average". It's much more complex than most people realize, as the following problem found in many an algebra text illustrates:

John travels 3 miles per hour from home to work and 4 miles per hour back home. If the entire round trip travel time is 1.5 hours, how far does John live from his business.

The following answer was given:

Distance = average speed x time, average speed is (3 + 4)/2 = 3.5 mph, so total distance must be
D = 3.5 miles/hr x 1.5 hr = 5.25 miles, hence John must live 5.25/2 = 2.625 miles from work.

Explain why this answer is wrong and find the correct answer to the problem. Hint: Look at this problem first.

In both these examples the "average" of the two speeds times the total time interval is 50 miles. But is either total distance traveled 50 miles? Do it in your head!

1 mph for 9 hours, then 9 mph for 1 hour
9 mph for 9 hours, then 1 mph for 1 hour
What about 4 mph for 5 hours then 6 mph for 5 hours, what's the difference?

Exercise 4: You are travelling in a car and the speedometer arrow is stationary at 60 mph. What information does it give you?

Suppose the speedometer reads 60 mph when you glance at it, but you notice it is moving to the left. What information does it give you?

Suppose the speedometer reads 60 mph when you glance at it, but you notice it is moving to the right. What information does it give you?

You make a bike trip to Dadeville (20 miles away) in one hour. What is your average velocity ? (magnitude and direction).
Suppose you would like to know how fast you were going when you reached the river bridge, (it's at the bottom of a hill), is this velocity a good approximation of that?

Suppose you obtained clock readings for reaching and leaving the bridge, and the bridges' length. Would the bridge length divided by the time interval it takes to cross the bridge be a better approximation of your speed when you reached the bridge? How could we make it better? Discuss.

New Concept: Instantaneous Velocity: How is it obtained and what does it mean?

- What's going on when the speedometer swings CW, CCW.
- What is the car doing at the instant you read the speedometer?

It's direction is the direction the object is travelling at this instant. It's magnitude is the limiting value of the ratio

as t gets closer and closer t_o. Or, speaking in the language of Calculus,
In other words, velocity is the derivative of displacement with respect to time.

OPERATIONAL DEFINITION of Instantaneous Velocity- It has 2 parts:

1. Its Direction is the direction of the tangent line to the displacement vs time curve at that point.

2. Magnitude- = = the slope of the line tangent to the displacement vs time curve at a point.

Interpretation of the Concept.
What does uniform velocity mean? - Tells us the object travels same displacement each second.

What does average velocity mean? - Tells uniform velocity object would have undergone in the same time interval, if it were travelling with constant velocity throughout the entire time interval.

What does Instantaneous velocity mean? - Uniform velocity object would continue to move at if its velocity became constant at this instant.

I want you to notice something else here. We have taken a really simple definition of velocity and broadened it considerably. We will continue to refine it into a more and more complex concept as we become more familiar with it.

The point is, this is a repeated characteristic of physics. As one is developing a model of a new concept in physics, one begins with the very simple, and as understanding broadens, the concept does too.

Speaking of which; let's try to draw displacement vs time and velocity vs time graphs for A,B,C & D in Exercises 1 and 2.

Acceleration is not about the present, it's about the future!:

Obviously the object D in exercise 2 above is accelerating. How do you know?

Yeah, it travels more distance in the next time interval than it did in the last. That makes sense. Velocity tells us how far an object travels on average in a given time interval, so a change in velocity should tell us how much more (or less) distance is traveled in the next interval than the last and in fact, this is the most useful description of acceleration to remember. To help visualize this, perform the following exercise.

Figure 1 below represents an object travelling with constant velocity. Each of the arrows represents the instantaneous velocity at the position rather than average velocity over the interval.

Now lets give this object a constant change in velocity Dv , which represents a fixed number of meters/second. On figure 2, attach one Dv1 to the velocity vector in the second interval, two in the third, etc.

That's positive acceleration!
Figure 3 is like Figure 1 only with higher constant speed.

Let's give this object a constant change in velocity of Dv₂ . Since it is in the opposite direction, the sum of v and Dv₂ will result in a shortening of v in each interval. So in Figure 4 shorten the second arrow by one Dv₂ , the third by two, etc.

That's negative acceleration!

Let's redraw a motion diagram for D using arrows whose length represent instantaneous velocity at the positions rather than average velocities over intervals and obtain a visual representation of acceleration .
Suppose we examine the ratio , where v₁ = v(t₁), and v₂ = v(t₂), both instantaneous, and t₂ is always the later time (> t₁).

-What information does it give us? It tells us how many more meters the object will travel in the next second, on average, than it did in the last second. This is the average acceleration. Notice that instantaneous velocities are required to find average acceleration.

Meaning? Suppose the object has velocity of 20 mi/hr and is accelerated uniformly at a rate of a = 5 mi/hr each second, what would be the velocity in 1 s, 2 s, 2.5 s, 3.5 s ?

Suppose an object has a velocity of 20 mi/hr and a = -4 mi/hr each second. What is its velocity in 1 s, 2 s, etc.. What is it doing?

Have you ever heard of the number 9.8 m/s₂, or 9.8 m/s/s ? What does it represent?

Suppose we drop a ball from a very high bridge. How fast will it be going after 1 s, 2 s, 4.5 s ?
Suppose we throw a ball down from a very high bridge at 20 m/s. How fast will it be going after 1 s, 2 s, etc. ?
Now suppose we throw a ball straight up from the edge of the bridge at 20 m/s. How fast will it be going after etc.?

So what does it mean on a motion diagram? Draw for each of the above situations and derive how to obtain average acceleration from two consecutive velocity vectors.

(It tells you how much to lengthen or shorten the velocity vector in a motion diagram in each time interval.

Ex. a. Draw a motion diagram of an object with positive velocity and positive acceleration.

b. Draw a motion diagram of an object with positive velocity and negative acceleration.

c. Draw a motion diagram of an object with negative velocity and positive acceleration.

d. Draw a motion diagram of an object with negative velocity and negative acceleration.

Discuss acceleration at top of flight in above bridge problems where initial velocity is upward.

Discuss the algebraic definition of instantaneous acceleration and its meaning.-

How could we describe this quantity as instantaneous change of velocity or instantaneous acceleartion? Take another limit! Namely,

. Can you describe its meaning?

The direction of an object's acceleration at some arbitrary time t can be dertermined if we know its velocity at two different times separated by a short time interval Dt . The procedure is outlined and illustrated below.

Original Velocity: Draw an arrow representing the velocity v of an object at time t.

New Velocity: Draw another arrow representing the velocity v' of the object a short time Dt later at time t'.

Velocity Change: To find the change in velocity Dv during the time interval Dt, place the tails of v and v' together. The change in velocity Dv is a vector that pionts from the head of v to the head of v'. Notice the figure at the right that v + Dv = v', or rearranging, v' - v = Dv. (That is, Dv is the change in velocity.)

Acceleration: The acceleration equals the velocity change Dv divided by the time interval Dt. needed for that changa;that is, a = Dv/t. If you do not know the time interval, you can at least determine the direction of the acceleration because it points in the same direction as Dv.