Motion Language
By Christopher MooreKinematics
Physicists have a fancy word that we like to use for the study of motion. We call the study of motion kinematics. The purpose of studing kinematics is to develop an understanding of how stuff moves. We see stuff move all of the time. Studing motion helps us to understand not only why objects move the way they do, but more importantly it allows us to develop models of that motion. With a good model we can predict how an object will move before it moves. Handy information if you want to throw a good pitch, or safely navigate a sharp turn.
Physics is a mathematical science. And that means that we’ll have to do some math. But throughout this tutorial I’ll try to link the concepts with the math. That is to say we’ll put the math into words. On this page, I’ll go through some definitions so that we are speaking the same language. Physicists have very specific words that we use to describe motion. On the next page we’ll connect these words with the math.
Scalars and Vectors
We’re going to talk about motion words such as distance, displacement, speed, velocity, and acceleration. These words describe mathematical quantities each of which falls into one of two categories: the quantity is either a vector or a scalar.
Scalar: Mathematical quantities that can be described only by their magnitude. Vector: Mathematical quantities that can be described by their magnitude and their direction.
Ok, so what’s the difference?
Try to imagine you are very hungry and you are in a strange town. You walk up to someone and ask were is the closest Burger King. The conversation could go like this:
You: Excuse me sir, were is the closest fast-food joint?
Stranger: There is a Burger King 2 miles away.
Our good stranger has told you that the Burger King is a distance of 2 miles from were you stand. He has given you the distance to the Burger King. So you walk merrily away dreaming of a whopper. Unfortunately you soon realize you have NO IDEA which way to walk! You’ve only been given a scalar. So now you have to stop someone else:
You: Excuse me sir, were is the Burger King?
Stranger: I just heard that other guy tell you 2 miles away. What else do you want?
You: Yes I know the Burger King is 2 miles away, but in which direction do I walk?
Stranger: Oh. Of course. Walk 2 miles to the north.
Now you can get to the Burger King. You have both the magnitude and the direction. This new stranger has given you the displacement of the Burger King from were you are. A vector.
So the distinction between vectors and scalars is important. Your lunch depends on it.
Distance vs. Displacement
There is a tendency to interchange the words distance and displacement. But as we just saw, they mean different things. If I tell you the displacement then I’m giving you more information than if I were to just tell you a distance.
Distance: How much ground has the object covered during its motion. Scalar.
Displacement: How far out of place is an object from were it started. Vector.
Most of us have a pretty good understanding of distance. But what about displacement? Let’s look at an example.
The bell rings. Jenny gets up from her desk and walks 10 meters down the hall to her next class. She then realizes that her next class is in the same classroom she just left (it’s been a long day, don’t ask how Jenny forgot this). So she turns around and walks 10 meters back to the same desk.
What distance did Jenny walk?
What is Jenny’s displacement?
It’s easy to see that Jenny covered 20 meters of ground. And from the picture, we can see that Jenny is in exactly the same place she began. So her displacement is ZERO! So if you thought that distance and displacement were the same thing, then you now know that you were very wrong.
Because displacement also involves direction, then we have to consider it as well. So all of the distance Jenny walked in one direction is canceled by the distance that she walked in the reverse direction.
To make sure you understand the difference between the two, I’m going to give you a little quiz. On a piece of paper, draw pictures that describe each of the following situations. Try to answer each question yourself before looking at the answer I’ve given.
Exercise 1: John walks east 5 meters, then turns north and walks 2 meters, then turns west and walks 5 meters, and finally turns south and walks 2 meters. What is John’s distance traveled and displacement?
Distance:
Displacement:
Exercise 2: Jimmy walks east 3 meters, then turns north and walks 3 meters, then turns west and walks 5 meters, and finally turns south and walks 3 meters. What is John’s distance traveled and displacement?
Distance:
Displacement:
Exercise 3: Elizabeth thinks all of these guys walking in circles is ridiculous! So she heads off to the gym going north. She arrives at the gym after walking 200 meters and then realizes that she left her gym bag in her car 100 meters to the east. She goes and gets her bag. Half way back to the gym her friend Jeff stops her to talk. He reminds her that he still has her iPod, so they go to Jeff’s car which is 100 meters east of Elizabeth’s. What is Elizabeths’s distance traveled and displacement?
Distance:
Displacement:
That last one was a little tricky, wasn’t it? For Exercise 3 you needed to use a little trigonometry to determine the displacement. But if you draw a good picture of the situation, then it turns out that it isn’t that hard. Because I know it was a little harder than the first two, we’ll go over this one. Here is the picture I drew:
The drawing on the left shows all of the movements Elizabeth made. The drawing on the right ignores that back-and-forth walking that she did. You can see that Elizabeth is some distance away from her starting point. But how far? To determine that we can use the Pythagorean Theorem to determine the magnitude of the displacement:
From the picture we see that Elizabeth is in a direction north and east of her starting point. So Elizabeth’s displacement is 283 meters northeast.
In later sections we’ll go over how to work with vectors more methodically. But for now what is important is that you understand the BIG conceptual difference between distance and displacement.
Speed and Velocity
So you now know how far and in what direction is the Burger King. Only it’s 9:50 pm and they close in 10 minutes! (Don’t ask me why all of these people are just walking around this evening.) You know you have to run 2 miles in 10 minutes to get there in time and get that Whopper you’ve been longing for. You’re going to have to move fast. And this brings us to the concepts of speed and velocity.
Speed: The rate at which an objects distance changes. Or more simply: How fast an object is moving. Scalar.
Velocity: The rate at which an objects displacement changes. Vector.
You now know there is a pretty big difference between distance and displacement. Well, there is a similar difference between speed and velocity. But before we get to that, let’s talk a little bit about what a “rate” is.
Rate: A ratio indicating a relationship between two different types of measurements.
In the case of speed and velocity, the measurements are either distance or displacement and time. So our definitions can be re-written:
Speed: How much ground an object covers in a certain amount of time.
Velocity: How much an objects position changes in a certain amount of time.
So you’re back on the street and you’re heading to Burger King. (I should be charging for such excellent product placement.) You know that in order to get that hamburger you have to go a distance of 2 miles in 10 minutes. So your speed is 2 miles per 10 minutes, or 0.2 miles per minute.
Before you set out, though, Jenny from above asks you where you are going and you tell them Burger King. She wishes to join you in your run for sweat soda pop and a hamburger. The two of you set off.
Unfortunately, half way there Jenny realizes she left her wallet at the point the two of you started your run. So she frantically runs back while you continue to head to the Burger King. The situation looks as follows:
You get to the Burger King in exactly 10 minutes, exactly the same amount of time Jenny takes to get her wallet.
What was your speed over the entire trip?
What was Jenny’s speed over the entire trip?
What was your velocity over the entire trip?
What was Jenny’s velocity over the entire trip?
Average Speed and Average Velocity
To get a little better understanding of the answers to the questions above, I’m going to introduce you to the mathematical definitions for average speed and average velocity. You’ll find that many times a physics idea can be expressed more easily in mathematical equations. Words can get cumbersome, and although they help to understand the concepts, thery’re lousy for solving problems.
The is the greek letter uppercase-Delta, and it means “change in”. So the first equation would read as follows: “The distance covered divided by the change in time”. And the second equation would read: “The change in position divided by the change in time.”
You may have also noticed that I have labeled speed as and velocity as
. The little arrow over a letters head means that it is a vector and has direction associated with it.
These are “averages” because during the course of any given trip, your speed and/or velocity can change. If you come to a stop sign while driving in your car, then your speed and velocity must change in order to avoid a collision or a ticket. But over the entire trip you have only one average speed and one average velocity.
For an example, let’s look at how I arrived at the answers to the questions about Jenny’s and your little run. The first question asks about your speed — so let’s compute your average speed. We know you traveled a total distance of 2 miles and you did this in a time of 10 minutes, so:
Similarily, we know that Jenny traveled a total distance of 2 miles in 10 minutes:
So both Jenny and you went an average speed of 0.2 miles/minute, even though you ended up with a full belly and Jenny did not. But what about each average displacement? Well, at the end of the 10 minutes, you were displaced 2 miles from your original starting point, so:
Jenny, on the other hand, was right back to were she started at the end of 10 minutes, so:
You can think of it this way: Jenny ran with a velocity of 0.2 miles/minute for 1 mile in the north direction, then she ran 0.2 miles/minute for 1 mile in the south direction. Her “north velocity” exactly cancels out her “south velocity!” Once again you should realize that you must be careful with words. Although Jenny had an average speed, her average velocity was equal to zero.
Instantaneous Speed and Instantaneous Velocity
As I said before, speed and velocity can change over the course of a trip, so this brings us to the concepts of instantaneous speed and instantaneous velocity. You can think of instantaneous speed as the number displayed on your speedometer in your car (or motorcycle, or tricycle if it has a speedometer.) It is the speed you happen to be going at any given instance in time. Your instantaneous velocity is similar, only it has a direction associated with it. So you cannot read instantaneous velocity off of a speedometer unless your speedometer is fancy and has a compass that indicates direction as well.
We express instantaneous speed and velocity mathematical like so:
The is the greek letter lowercase-delta, and it means “an itty-bitty change in”. Because we are concerned about how fast we are going at a specific instance, then we compute the very small distance traveled over a very small period of time. Unfortunately, to accurately do calculations with
s, you have to do calculus. I’ll go over a little bit of calculus later on (don’t worry too much. It will be painless.) We’ll also look at approximating instantaneous speed and velocity.
If we go back to the example of Hungry Jenny, then we can see that although she has ZERO average velocity, her instantaneous velocity at different points during the trip was definitely not zero. Let’s look at 3 different points during the trip: t1=2.5 mintues, t2=5 mintues, and t3=7.5 mintues.
See if you can answer the following questions:
What was Jenny’s instantaneous velocity at t1?
What was Jenny’s instantaneous velocity at t2?
What was Jenny’s instantaneous velocity at t3?
To make sure you understand the difference between speed and velocity, I’m going to give you another little quiz. On a piece of paper, draw pictures that describe each of the following situations. Try to answer each question yourself before looking at the answer I’ve given.
Exercise 4: Ricardo has a day of fun in the sun planned. He leaves Richmond, VA at 8:30 am heading towards Virginia Beach, which is 108 miles away directly East. He arrives at 10:30 am. What was Ricardo’s average speed? What was his average veolicty?
What was Ricardo’s average speed?
What was Ricardo’s average velocity?
Exercise 5: Upon arriving in Virginia Beach, Richardo’s friend John (who cam along for the ride) notices that it’s really cloudy. They decide that instead of lying on the beach, the would rather ride amusement park rides. So they turn around and head to Busch Gardens in Williamsburg, VA, which is 58 miles directly West. They arrive at 11:30 am. What was Ricardo’s average speed and velocity for the entire trip, starting in Richmond?
What was Ricardo’s average speed?
What was Ricardo’s average velocity?
Exercise 6: Kelly has to drive from Richmond, VA to Washington, DC, which is 97.5 miles North up Interstate 95. Traffic is very light, so she can drive the speed limit the entire way (65 miles per hour, Kelly always obeys the speed limit 
). The meeting she has to attend starts at 3:00 pm. What time does Kelly need to leave Richmond in order to arrive on time?
Acceleration
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I have a question about the problem where Ricardo and his friend turn around at Virginia Beach at approximately 10:30a.m. and travel 58 miles West to Busch Gardens arriving at 11:30a.m.. I understand how the average velocity was calculated but it stated that their average speed was 55.3m.p.h.. Did they not travel 58 miles in a given hour? If so why wasn’t their average speed calculated as 58m.p.h.? 58 miles in one hour. 58m.p.h. Thanks
Comment by Ken Moore — Friday -- October 19th, 2007 @ 3:37 pm