Describing Motion: Velocity

Position and Distance Motion is all around us and in us. Some examples: even while standing motionless the Earth whips us around as it rotates around its axis of rotation; we take a headache tablet and our blood carries the medicine at a certain velocity through our circulatory system; we speak and it takes a certain time for our voice to reach the listener; we stub our toe and our nerves transmit the electrical pain impulses at a certain speed to our brain.

An important part of talking about motion is being able to describe where an object is located at a certain time. In other words, you want to be able to state the object's position. If you want to give accurate information about an object's location or position, you must compare where it is to some reference point. For example, if your car is broken down, you might be able to use a cell phone to tell your parents you are on Highway 13, 5 km East past the "Dollar Value" convenience store. In this example, the reference point is the convenience store. Here's another example: you walk 10 steps E of the blocks in the school foyer. What is your position? Easy - its 10 steps E of the blocks in the school foyer. Again, see how we describe the object's position in terms of its distance and direction from a reference point, the blocks in the foyer in this example. (position = distance + direction from ref. pt.)

As an object moves, it creates a trail called the frame of reference. In the broken down car example above, the frame of reference is just the path of the car from the convenience store to the spot where it broke down. When you choose the convenience store as the reference point, you are saying the store is the starting point in the car's frame of reference. Insta-revu: i) As the car moves, it creates its frame of reference. ii) The car's motion is compared to the reference point, location of the store.

Frames of reference are all around us. Think about a jetplane flying across the blue sky - its white exhaust trail (called the contrail) is its frame of reference. Did you ever notice the slime trail left by a slug poking along across the grass? That is its frame of reference. Imagine this: you're walking along a wet beach. Your first step in the wet sand is the reference point, your last step in the sand is the finish and your string of footsteps is frame of reference. Can you always see the frame of reference? No. Think about walking down a hallway here at school; you could not see your steps (your frame of reference) in this situation.

Sports is a situation where knowing about reference points and frames of reference is very important for the players and to the coach. Think about football - each player is supposed to be in a certain location compared to a certain opponent (a reference point) and each player is supposed to run in a certain path (run in a certain frame of reference).

Vectors and Scalars A vector is a type of value that tells the distance and direction an object has moved. If you say, "I walked 0.4 km east," you are giving a vector value because you have told how far you walked and the direction in which you walked. In the broken down car example above, what was the position of the car? It was located 5 km E past the convenience store. Is position a vector? Yes, your position information about the car tells the distance and direction the car moved past the reference point (the convenience store). So, position is a vector quantity because it contains distance and direction.

Using these ideas about position, what is a simple definition for it? Look at this one: position is the separation (the distance and the direction) between the reference point and the object.

A scalar is a value that tells only how far an object has gone. For example, you say, "Today, I ran 2.3 km." Did you tell how far? Yes. In what direction? No, so you are giving a scalar value. Does a scalar need a reference point? No, but your scalar value might mention it. You could say, "Today, I ran 1.7 km past the Post Office." You are still telling only the distance you ran but not the direction so you are still just giving a scalar value.

Q. You are a member of a search team. Describe the search pattern you would create for each of the

following situations. Use the ideas of vectors and scalars.

a) A missing person left camp on an ATV with an unknown amount of gas.

b) A missing person left camp on an ATV with gas for a 15 km trip.

c) A missing person left camp heading E on an ATV with an unknown amount of gas.

d) A missing person left camp heading E on an ATV with enough gas for a 15 km trip.

Displacement and Average Velocity As an object moves, it creates a continuous sequence of "time-position" values. Think of yourself walking across a wet lawn; each footstep is made at certain time and in a certain spot on the lawn. Each step can be called a time-position value. Think of the trail left by a rabbit hopping across a snowy field: each footprint is an instantaneous position and corresponds to a certain time value. Of course, an object might move and leave no obvious trail but there would still be an invisible line of time-position values.

As an object moves along, its position keeps changing of course. The object's displacement is just its change in position as it moves from point A to point B. And, because position is a vector, so is its displacement. We could say an elephant sauntered across the African grassland from a spot 50 m East of a waterhole to a spot 300 m East of the same waterhole. Since we have information about its two positions, we can know its displacement. Just subtract the two positions, i.e., 300 m East - 50 m East = 250 m East.

Finally, the object's average velocity is the ratio of its displacement changes to its corresponding time changes, i.e., how far it moves in what time. For example, if the elephant took nine minutes to walk the 250 m, find its average velocity. Use v_AVG = d/ t = 250 m / 540 seconds = 0.46 m/s. And, if we know the average velocity, we can find the time it takes an object to travel a certain distance. For example, how long would it take the elephant to walk 750 m? Use t = d / v = 750 m / 0.46 m/s = 1630.4 s.

A. Time-Position Graphs

Visuals of all sorts are a powerful, sometimes instant way of learning and graphs are a common type of visual to help you investigate and describe motion. A time- position graph shows where an object is located at various times. Just looking at a time-position graph gives a general, qualitative idea of the speed and direction of an object. Calculating the slope gives a more accurate, quantitative, idea about the object's velocity. (qualitative = non-numerical, e.,g, the cat is running quickly; quantitative = numerical, e.g., the dog is trotting along at 0.4 m/s)

A KEY !!! part of graphing skills is working with SLOPE. To find it, just use: m = y / x or,

m = y2 - y1 / x2 - x1. But, what does the slope mean? In a general way, all the slope does is compare how much the y values and the x values are changing. In other words, the slope just tells us how quickly a situation is changing. For example, the slope of a time-temperature graph would tells us how quickly the temperature rises or falls during the day. The slope of a distance-fuel level graph would tell an engine technician how quickly a new engine design was using gasoline in test drives. The slope of a food amount- weight gain graph would tell a nutritionist if a diet is healthy. So, do you see the meaning of the slope depends on what the graph is about? The meaning of the slope depends on what info is on the x and y axes.

The motion graphs we're looking at have time info on the x axis and position info on the y axis. So, when we find the slope of a t-p graph we are comparing position info with time info. In other words, the slope of a t-p graph is found by: m = p2 - p1 / t2 - t1. We are calculating how quickly an object's position changes as the time ticks by. And, comparing position (in meters) with time (in seconds) is the same as finding the speed of an object! Another way of seeing the slope of a t-p graph means speed is by looking at the unit of the slope value. Since the slope is found by position / time, the slope's unit will be m / s and this unit means speed!! (LOOK: the slope of a t-p graph = the object's speed.)

A COMMON mistake in graphing is getting the axes reversed - easy to do but actually a big mistake. The time values are on the x axis and the position values are on the y axis. LOOK: remember the popular watch brand Timex; time goes on the x axis!!

We all know what a map is even if we are not all so clever at using one. East is to the right and West is to the left. All the towns and cities are small dots or circles. But, a t-p graph is NOT a map, so to reduce confusion, please take note of the following points about t-p graphs:

i) any plot line that rises means the object is moving in an easterly direction.

ii) any plot line that stops above the x axis means the object's position is E (right).

iii) any line that falls means the object is moving in an westerly direction.

iv) any plot that stops below the x axis means the object's position is W (left).

v) the entire length of the x axis, all the way over, represents just one spot, the reference point, the start!

No motion A time-position graph for "no motion"

is ANY straight horizontal (flat) line. Where the

line intersects the y axis shows us where the object

was resting and continues to rest. If the plot line

joins the y axis above the x axis, the object was

resting in the E. If the plot line joins the y axis

below the x axis, the object was resting in the W.

If the plot line is on the x axis, the object was

resting at the reference point.

Because the object is motionless, its change of position ( p) is 0. And, if the p is 0, the slope will be 0. How could you show this situation? Simple, just stand still.(LOOK: no motion = no speed = a flat line = slope of 0.)

Constant velocity The time-position graph describing the motion of an object moving at a constant (uniform) velocity is a straight rising or falling line. Where the line intersects the y axis shows us where the object was resting before it started to move. If the plot line joins the y axis above the x axis, the object was resting in the E. If the plot line joins the y axis below the x axis, the object was resting in the W. If the plot line is on the x axis, the object was resting at the reference point. These straight line plots have many variations. How would you show constant velocity motion? Simple, just walk with an even stride, an even speed.

LOOK: i) a rising plot means the object is moving eastward (right); slope is +

ii) a falling line means the object is moving westward (left); slope is -

iii) a steep up or down line means rapid motion; slope is large + or -

iv) a shallow (nearly flat) up or down line means slow motion; slope is small + or -

v) if a plot line stops above the x axis, the object finishes in the E

vi) if the plot line stops below the x axis, the object finishes in the W.

Uniformly changing velocity When an object accelerates, it is increasing its speed. An object accelerating evenly to the E creates a plot that is begins flat and then gets steeper and steeper. Smooth acceleration to the W creates a downward arcing plot, just the mirror image. How would you show this motion? Simple, just walk faster and faster. (LOOK: acceleration = flat steep)

A smoothly decelerating object moving to the E creates an upward plot that is steep to begin with but gets flatter and flatter. An object decelerating evenly to the W creates causes a downward plot that is steep to begin with but becomes flatter and flatter. How would you show this motion? Simple, just walk slower and slower. (LOOK: deceleration = steep flat)

What about slopes for these acceleration or deceleration time- position graphs? We need to get into a topic called instantaneous velocity. The t-p graphs for constant velocity motion are all straight line plots and a straight line has one slope value. (LOOK: constant speed = one speed = a straight plot line = one slope value) But, if the object's speed is changing, we say it has an different instantaneous velocity at each instant of its motion. The object's plot is a curving line and, a curving line has a different slope value at every spot along its length (because the speed is different at every spot)!

Think: it makes no sense to talk about "the" speed of an object that accelerating or decelerating like it has just one speed. It makes perfect sense to ask about the speed of the object at a certain time. So, how do we find the speed of an object a particular time? In other words, how do we find the slope of a curving line at a certain time? We must use a graphing skill from math called finding the "slope of a tangent". To find the object's velocity at a certain time in its motion, find the specified time on the x axis, go up to the curving plot line, draw a tangent at that spot on the plot line, and find the slope of the tangent. See below for some examples of instantaneous velocities from uniformly changing velocity t-p graphs.

B. Time-Velocity Graphs

Although one can calculate an object's velocity by finding the slope of a time-position graphs, if we have a time-velocity graph, we can just look at it to see the object's speed. Because its easy to confuse t-p and t-v graphs (they have different meanings), attention to situations and axis labels is key.

No motion The plot is horizontal (flat)

and lies exactly on the x axis itself

because the speed = 0 there.

Uniform velocity The plot is a straight

horizontal line intersecting the y axis at

the velocity value, + or - depending on

direction of motion. (LOOK: they look like t-p graphs for "no motion" but don't mean the same thing.)

Note: to find how far (displacement) an object with uniform velocity has travelled in a given time span, look at the two times, draw a rectangle under the t-v plot and, find its area. (LOOK: slope of t-p graph = velocity and area of t-v graph = position)

Uniformly changing velocity These graphs

have straight rising or falling plot lines.

(LOOK: they look like t-p graphs for "uniform velocity" but don't mean the same thing.)