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.

Relative Motion            Imagine we are sitting on our porch steps looking at the world pass by. When a person walks past we get a sense of their motion and it is often easier to judge slow motion than fast motion. But, what if we are moving at the same time the other person is? Then, we are in a relative motion situation, a situation of “optical illusion”.

             Racecams are popular with television viewers because they can carry the viewer inside the action. When the racecam is panning around, we see the speed of the other cars relative to that in which the racecam is located. Even though the actual speed of the vehicles may be close to 200 mph, they are hardly moving relative to each other; their relative speeds are small. Over and over, the cars slowly approach a bit then fall back. Sometimes they keep exact pace with each other. However, if something causes one of the vehicles to suddenly veer sideways, we quickly get a sense of its actual speed as it rockets away from the racecam. Below are some common situations. The observer’s viewpoint is is key.

Q. a) You are standing on an escalator moving at 0.6 m/s. What is your speed relative to the escalator?

         Just look down. Are you moving? No, so your speed is 0 m/s.

     b) What is the escalator’s speed relative to a person standing, just looking at the escalator? Since it is

         going at 0.6 m/s and they are going at 0 m/s, the escalator is moving at 0.6 m/s.

     c) The person sees you moving at what speed? Since you are just letting the escalator carry you along

         at 0.6 m/s, that is how fast the person sees you moving.

Q. a) You are walking ahead at 0.2 m/s on the escalator already moving at 0.6 m/s. What is your speed

         relative to the escalator? Just look down. You steps move you along at 0.2 m/s so, that is your

         relative speed.

     b) The person standing watching the escalator sees you moving at what speed? Since you are walking

         0.2 m/s along the escalator that is already moving at 0.6 m/s, the person sees your combined speed

         of 0.8 m/s.

Q. A car moving at 45 kph is approaches your car moving at 55 kph. What speed do you see for the

     other car? You see the combined speed of 100 kph. This is why head-on impacts are so energetic– the

     energies of the two objects combine. Note: each driver has the same sense of the combined speed.

Q. a) A car “A” moving at 60 kph passes and pulls away from a car “B” moving at 40 kph. What speed

         does B see for A as it approaches? Just find at the difference: 60 – 40 = 20 kph. B sees A coming up

         at the relative speed of 20 kph.

     b) What about when A has passed and is pulling ahead? The relative speed is the same, 20 kph, but

         now B would see A is pulling away at 20 kph, not its actual speed of 60 kph.

Q. a) A jet plane moving at 600 kph fires a rocket that moves at 900 kph. To the pilot, how fast is the

         rocket going? The pilot doesn’t notice the speed of her own plane, so she sees the rocket going

         just its own speed, 900 kph.

     b) What is the speed of the rocket to ground based radar? Because the radar is “outside” the jet, it

          sees the rocket’s combined speed of 1 500 kph (600 + 900 kph).

     c) How fast does the rocket seem to be moving to another jet trying to escape at 700 kph? With 3

         objects, the situation needs just a bit more thought. Because the pilot of the escaping jet is “outside”

         the first jet, he could be expected to see the rocket’s combined speed of 1 500 kph (600 + 900 kph).

         But, like one car passing another (above), we must find the difference between the rocket’s

         combined speed and that of the escaping plane: 1 500 kph – 700 kph = 800 kph. the escaping pilot

         sees the rocket catching up at 800 kph.

     d) If the escaping pilot swerves and the rocket misses and pulls ahead, what will be its speed to the

         escaping pilot? Just like the driver of the overtaken car (above), the escaping pilot sees the rocket

         pulling away at 800 kph.

     e) How fast will the rocket be going to a jet approaching at 550 kph. Easy, just like the approaching

         cars, combine all the speeds: 600 + 900 + 550 kph = 2 050 kph.

             Another type of relative motion is based on different start times and different speeds, not different points of view. A slow object starts out and after a delay, a faster object starts. The point of the question is to find the chase down time and distance.

Q. a) The tortoise takes off at 0.2 m/s and, after a delay of 0.5 days, the hare starts out at 1.3 m/s. From

         the time it starts, how long does it take the hare to chase down the tortoise? At the spot where the

         fast object overtakes the slower object, they are both the same distance from the start. The skills we

         are using are simultaneous equations followed by expansion, gathering similar terms, division to

         find the chase down time.

d fast = d slow

vt = vt

1.3 m/s t = 0.2 m/s (t + delay time)

1.3 m/s t = 0.2 m/s (t + 43 200 secs) (0.5 d * 24 hr/d * 60 min/hr * 60 s/min)

                                  1.3 m/s t = 0.2 m/s t + 8 640 m

               1.3 m/s t – 0.2 m/s t = 8 640 m

                                  1.1 m/s t = 8 640 m

                                               t = 8 640 m / 1.1 m/s

                                               t = 7 854.5 s

     b) The hare reaches the tortoise at what distance from the start? Use the chase down time to find the

         chase down distance, ie. substitute the t into the d. Altogether, the math sequence is: Egads or

         Expand, gather, divide, substitute.

                                        d fast = v t

                                                 = 1.3 m/s * 7 854.5 s

                                                 = 10 210.8 m.

       As a work check, find the d slow to see if it equals the d fast (as it is supposed to).

                                       d slow = 0.2 m/s (t + delay time)

                                                  = 0.2 m/s (t + 43 200 secs)

                                                  = 0.2 m/s (7 854.5 s + 43 200 s)

                                                  = 0.2 m/s * 51 054.5 s

                                                  = 10 210.9 m (good agreement!)

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 = Δ d/Δ 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

Graphical Analysis of Motion

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 (Δ p) is 0. And, if the Δ 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.

Here are some examples of constant velocity time–position graphs.

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.

             Recall that a curving line is made up of an infinite number of tiny straight lines meeting end to end at slightly different angles. It is the difference in the segments’ meeting angles that creates the overall curvature. Drawing a tangent is really just selecting one of the tiny segments and extending it into a straight line of useable length. See below for some examples of instantaneous velocities from uniformly changing velocity t–p graphs.

Variably changing velocity When a car is under uneven braking (you push down suddenly on the brake pedal) or acceleration (you suddenly jam down the gas pedal), the t–p plots will be more variable and perhaps extreme than the smooth t–p arcs for Uniformly changing velocity.

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

but lies 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.)

Variably changing velocity The graphs look like time–position graphs for Uniformly changing velocity with their smoothly sloping arcs.

Applications of Velocity           A love of speed is common among humans. Through the centuries, man has competed on foot, on the backs of animals and in mechanical contraptions. For the past forty years, vehicles have been specially designed to race along railway track, across land and over water in search of new speed records.

             Still, there are some who prefer peace and quiet, a slow measured pace. But even when we are supposedly standing still everything is always moving at incredible rates: the earth is spinning on its axis at 1 600 kph. And, the earth revolves around the sun at ; the solar system moves through its Galaxy, the Milky Way, at ; the Milky Way galaxy revolves at ; the Galaxies are racing away from each other at !

Q. Find the speeds of the following motions: a) the earth’s revolution b) the earth’s orbit

     c) our solar system’s orbit d) the expansion of the universe.