purpose - ket virtual physics labs … · web viewstl-velocity-03-27-key.docx/15print date: ... ...

Name School ____________________________________ Date

STL2 Class Activities for 1-D KinematicsPurpose

Learn the use of some of the tools, techniques, and terminology related to the Dynamics Track and Grapher. Learn to use good lab measurement and graphing techniques. Learn to determine the displacement, speed, velocity of an object moving at a constant velocity using motion

diagrams, graphs, equations, and vectors. Learn to interpret position vs. time and velocity vs. time graphs.

EquipmentDynamics Track VPL Grapher PENCIL

Watch the Video (no audio) Overview at: http://virtuallabs.ket.org/physics/apparatus/02_kinematics/

Figure 1 – The Dynamics Track

document.docx /16 Print Date: 3/30/2017 2:59:00 PM

http://virtuallabs.ket.org/physics/apparatus/02_kinematics/

A. Explore the Apparatus; learn some things that you already know.Note: Brackets around a button or command name means to do it/click it. The same with a bar through it means turn it off.

Ex. [Go→] = Click Go→ [ ] = click Ruler icon (bottom of screen), [ ] = (Click the icon again to hide it.)

1. Start up the dynamics track. Don’t change any settings until you’re asked to. (Later you can do as you please. You can’t break anything.) Drag the cart around. Toss it. How do you stop it?

How could you specify where the cart is at a given instant – its position? We need numbers! Measurements! Also, the cart is pretty big. How might you specify where it is?

Could you be more precise about its location? Right-click-Zoom In (Ctrl-Zoom In) might help. Zoom Out or Show All when you’re finished. It’s actually a bit wonky at the bumpers. That’s OK. It makes it a bit more realistic.

2. Drag the cart all the way to the left. What’s its position now?

You should always measure as many digits as you can with certainty and then estimate one more digit. All of these digits, including the estimated ones, are referred to as significant digits.

3. How about if it was to the left of the ruler? Above? Behind? How might Amazon-like warehouses use this technique?

This chapter is about motion in one dimension, 1-D kinematics. We’ll stay on one axis, usually the horizontal, x-axis.

[ ]Hide the ruler.)

4. Move cart to left end. [Go→] (This means click the Go→ button.) Watch it for a few passes. What could you say about the motion of the cart?

Why is it jerky?

5. Click the cart to stop it. [Go→] from a stationary start at the left end and let it bounce once off of each end of the track. What do you think its three speeds were? (Note the Vo next to Go→ which should still be 100 cm/s.) Explain.

We’ll now explore several ways of determining velocity using various tools and strategies. This will help you learn to the apparatus in creative ways and to use a number of mathematical techniques that we’ll rely on throughout the course.


B. Determining the Speed of the Cart using Motion Diagrams

We’ve been using the term speed to refer to how fast the cart was moving. We define the average speed of an object as the ratio of the distance traveled over the time taken, the time interval. If speed is constant, speed = average speed.

Average speed, speed= distancetimeinterval , t

(1)

You’ve found that using [Go→] gives the cart a speed of Vo and that Recoil reduces the speed by a certain factor when the cart bounces off the bumpers. How can we verify this using Equation 1? All we have is a moving cart. We need to do some actual measurements of distance and time. One way is to use the Motion Diagrams tool which provides a logical visual representation of the cart’s motion.

1. Set Vo to 60 cm/s. Set Recoil to .5. Put the cart at left end of the track. [Set θ = 0°] and leave it there for this activity.

[○ Motion Diagrams] to turn on Motion Diagrams. Set Inkjet frequency to 5 dots/sec.(FYI later: To hide the Motion Diagrams just click the No Gates radio button [○ No Gates]. Not now.)

[Start Motion Diagrams] and then [Go→]

Allow the cart to pass the middle of the track three times. (Right-Left-Right.) Then [Stop Motion Diagrams] and stop the cart. Feel free to repeat this until you get a good run with three rows of dots and feel clear about what’s going on with the cart and the dots.

You’ve seen this motion before, but now you’ve got lots o’ dots to help you determine the cart’s speed. You need a distance and the time taken to travel that distance. Each dot is a record of the position of the cart (mast) at a given instant. We need to pick a couple of dots and find the distance between them and the time the cart takes to travel this distance.

[ ] and drag the ruler just below the top row of dots. You can also move the dots printout with [Feed Up/Dn]. The first 2 dots after launch and after bouncing at the ends are unreliable, so we don’t usually use them.

Let’s use Figure 2 for this example rather than the results on your apparatus. The explosion icon overlapping the first dot indicates that the cart was launched with Go→. Our first reliable dot would be the one just past 50 cm. What’s the position of that dot? The hatch mark to its left is 50.0 cm. The hatch mark to its right is 52.0 cm since the hatch marks are 2.0 cm apart. If it was exactly between them, it would be at 51.0 cm. But it’s a bit to the right so it might be 51.1 or 51.2 cm. There is uncertainty and there always will be with any measurement. This is not a bad thing. This is life. Knowing how to deal with uncertainty is good. One way we do this is by indicating the amount of uncertainty in a measurement.

Whenever you make a measurement, always record all the certain digits plus one estimated digit.These would be 51 and .1 or .2 respectively in this example, so you would record 51.1 cm or 51.2 cm.

You’re the judge of that last digit. What if it’s zero? Could you just leave it out? No. 52.0 cm means that you measured, including your estimation, to .1 cm and the estimation was zero tenths of a centimeter. 52 cm means that the 2 was the estimated digit. That would not be incorrect.

Figure 2: Draggable Ruler with Motion Diagram

2. Back to calculating the speed of the cart. We need to pick some distance, d, traveled by the cart, and the time interval, Δt it took to travel that distance. We could find the distance between any two dots we like. How about two adjacent dots? Or maybe two spaced very far apart? Which would be better?


The dot spacing is a bit irregular. If we used just a pair of adjacent dots, they are most likely closer together or farther apart than average. Which brings us to the obvious answer: Measure over a lot of dots so as to average out the variation. In between the first and last reliable dots there are short spans and long ones which tend to balance each other.

We’ll want to use a long distance. How about the third dot and the third to last dot.

d = large (right) position - small (left) position = cm - cm = cm

Now, we need a time interval, t, the time the cart takes to travel between our chosen points.

3. What does the inkjet frequency refer to? Feel free to experiment. Just reset it to 5 when you’re finished.

4. What does the value “T = 0.200 s” indicate? It’s called the period of the dots or of the ink jet that makes them. (Similarly, a pendulum, sound wave, the Earth, and many other things have periods. Nature likes cycles.)

Here’s an example from Figure 3, of how to use this information to find the average speed of the cart on the trip to the right. The launch and reflection dots have been removed in Figures 3 and 4, so all the dots are good for our use.

speed = (distance between two chosen dots) / (time interval)

distance = |position of the last dot – position of dot of the first dot| (Note absolute value. Distance has no direction.)

time interval = time when the last dot was created – time when the first dot was created

The time interval is a bit tricky since we don’t know any actual times. But we know the time interval between dots is .200 s. And we can easily determine the number of intervals by multiplying the number of dots, (counting the start and end dots) – 1 times the period, T. Here’s the result.

speed1 = (179.2 - 34.1) cm / ((13 – 1) × .200 s) = 60.5 cm/s

Figure 3

6. Take all the data you need from Figure 4 to calculate the speed of the trip to the left, speed2. Show your calculations below. Again, assume that all dots are good data. (You don’t have to use them all. But more is better.)

Figure 4

speed2 = cm/s

7. How do these numbers look? Are they incorrect? You should get very close to the 60 cm/s and 30 cm/s we expected. This is measured data. We should always assume that they will vary somewhat from what was expected, because these are measured data which include uncertainty.

Incidentally, counting is another matter. If you count the number of marbles in a jar and get the wrong number you’ve made a mistake, not encountered uncertainty.


C. Determining Velocity using Motion Diagrams and the Motion Sensor; Position vs. Time GraphsWe’ve been calculating the speed by calculating distance/time. This calculation ignores the fact that the cart is changing direction. Speed is a scalar quantity, which means that it has a magnitude, but no direction. We’re usually more interested in the velocity of an object. Velocity is a vector quantity which means it has both a magnitude and a direction. The direction of the velocity is just the direction of the motion. (The previous paragraph was way more important than it looked. We handle scalars and vectors very differently. (Sort of like cats vs. dogs at the vet.) We are mainly interested in vectors.)

Average velocity, v=displacement , Δxtimeinterval , Δt = change∈ postion , Δx

time interval , Δt=

x f −xo

t f−t o(2)

The initial position, xo, and the final position, xf, correspond to our earlier and later dot positions. We’ll often use numbers for subscripts instead of letters as shown in Figure 5. But the earlier x and t will always be subtracted from the later x and t.

The displacement, Δx is a vector from the starting point to the ending point of a given motion. Its magnitude is just the straight-line distance between the two points, and its direction is the direction from the starting point to the ending point as shown in Figure 5. We frequently use arrows to graphically represent vectors. The magnitude of a vector is indicated by the length the arrow and the direction is indicated by the arrow head on the vector. We often use the terms “tail” and “head” for the two ends of a vector arrow. And we usually just call the arrow itself a vector. (Pre-med types: Google “vectors medicine”)

Figure 5: A Displacement Vector

Let’s find the three velocities of our cart using the definition above. Since these three velocities are constant, we’ll just drop the term “average” for now. We’ll need three displacements and the corresponding time intervals. Thus, we need three (to, xo), (tf, xf) pairs. This time we’ll use a different tool to collect our data.

1. Run a trial with the same settings as before, (Vo = 60 cm/s, Recoil = .5) but this time instead of the sequence

[Start Motion Diagrams] [Go→], you’ll use [Sensor] [Start Motion Diagrams] [Go→]

[Sensor] refers to the Sensor at the left end of the track. This tool will collect our (time, position) data for us using the time for the ultrasonic sound waves it emits to reflect off the cart and return to the Sensor. Try it. It’s an on/off switch. Click to start and click to stop. You’ll also need to stop the motion diagrams, so use [Sensor] [Stop Motion Diagrams]

2. The graph ran out of room! We need more time. Change Tmax(s) to 25 s and try again. This time [Sensor] [Stop Motion Diagrams] just before the cart hits the right end for the second time. You should have something like Figure 6 and three rows of dots above the track.


Figure 6: Sensor (with data) + Motion Diagrams


Hopefully you can see how the graph will come into play later, but for now, let’s have a look at our data table. The first column contains gradually increasing times while the second column shows a constant position. What gives?

You’ll use the data table scroll bar and scroll track and the up/down arrows to move through the data in the scrolling table. Clicking in the scroll track above or below the scroll bar is very helpful for jumping one page of data at a time. If you drag the scrollbar downward, you’ll see the entire collection of data fly by. But there’s a lot of it. For better control use the scroll track. Click and hold to jump quickly.

3. From the table, what’s the maximum position value, including units? The minimum? What was the position value when you turned off the sensor? You’ll see that the maximum position isn’t likely to be found at the end of the list. Consulting the graph is helpful.

Let’s put our equation for average velocity to use. [ ] if the ruler is not already visible.

4. Your cart had three constant velocities which we’ll now calculate, this time using our data table. Set Match Graphs to 1. Drag the cart from side to side. What’s the connection between the location of the cart on the track and the location of the red dot on the vertical, position axis? (See Figure 7 for the next several steps.)

Note that the dependent variable, the position in this case, is always plotted on the vertical, y-axis, the ordinate.The independent variable, the time in this case, is always plotted on the horizontal, x-axis, the abscissa. Note that these standard names – x, and y would be replaced in this case by t, and x for time and position respectively.

5. Drag the cart to x = .40 m. Use the ruler and/or red dot to assist you. At the top right corner of the graph you see two rectangles. Drag the horizontal one down until the thin horizontal grid line passes through the red dot.

6. Repeat for x = 1.20 m.

Figure 7Notice that the grid lines actually pass through each of the three lines on the graph. Why?


7. Let’s find the velocity for the first trip from left to right. To use Equation 2, you need the time and position for two points on the first, rising line. We’ll use the two points where your horizontal grid lines crossed the first rising line. Let’s call the first point (t1, x1) and the second point (t2, x2). You’ll find each time and matching position by examining the data table.

Starting with the data table scroll bar at the top, click below it in the scroll track several times until you see changing position values indicating that the cart has started to move. Then use either of the other methods (probably the arrows) to move the bar until you see the position value nearest to .40 m. Finding a data point for exactly x = .40 m is unlikely. Just use the one closest to .40 m. Sample (t1, x1) values are provided in the left column below. Record your values in the blanks below them in the table. Repeat for the first time the cart reaches about 1.20 m. Record your values as before.

Your Δx values, that is x2 – x1 should be close to .80 m. Your time values will be different from the samples since you’d certainly click Go→ at a different time than the sample start time, but your Δt = t2 – t1 values should be close to 1.33 s.

8. Show the calculations of your average velocity, va, in the right column below the sample calculations. If your va result isn’t approximately .60 m/s see if you can find your error. Then finish up the table by measuring the other four (t, x) values in the table and calculating vb, and vc. You should already know what to expect the v’s should be.

t1 = 2.063 m x1 = .396 mt2 = 3.406 m x2 = 1.205 mt1 = m x1 = mt2 = m x2 = m

va=Δ xΔ t

=x2−x1

t 2−t 1=1.205 m−.396

3.406−2.063= .60 m/s

t3 = m x3 = mt4 = m x4 = mt3 = m x3 = mt4 = m x4 = m

vb=Δ xΔ t

=x4−x3

t4−t 3= .398m−1.196

9.719−7.031

t5 = m x5 = mt6 = m x6 = mt5 = m x5 = mt6 = m x6 = m

vc=Δ xΔt

=x6−x5

t6−t5= 1.201 m−.400

18.250−12.938

9. What does the negative sign on your second velocity, vb indicate? How did it get this sign? Use the term displacement in your answer.

10. What was the speed of the cart for this trip to the left?

11. In the space below, draw a scaled vector arrow for each of the three displacements. Label them Δxa, Δxb, and Δxc.

12. Compare and contrast the displacement vectors for the three trips we measured.

Note: vectors are usually indicated by bold type or with arrows above the name. Ex. Δx or Δ⃑ x. For kinematics, we traditionally omit both of these indicators except in hand writing where the arrows are often used. Go figure.

Speed: distance/time interval. (There is no standard letter for speed. Fortunately, we mainly work with velocity.)


Velocity, v: change in position/time interval = displacement/time interval = Δx/Δt = (xf – xo) / (tf – to)

D. The Slope of a Position vs. Time Graph for Motion with a Constant Velocity; Velocity vs. Time Graphs1. Run a trial with these settings: Cart at the left end. Vo = 60 cm/s, Recoil = .5, Tmax(s) = 10 s. [○ No Gates]

Here’s a new trick in case you’re interested. Click next to Vo. Watch what happens. It’s just [Go→] after a 5-s delay! So, if you like, whenever you see [Sensor] [Go→], you can instead use [ ] and then just before midnight, [Sensor].

[Sensor] [Go→] or [ ] wait [Sensor]. You should get something like Figure 8a.

Figure 8a

As you’ve just learned, this graph contains all the data we used in calculating va (right motion) and vb (left motion) along with any other (t, x) data we might need. But last time, we immediately looked up the data in the data table. So, what’s the big deal about having a graph? For one thing, a continuous set of data is available instead of just periodic readings.? And as you’ll soon see, a graph represents the motion in a visual way that a data table can’t. You can literally see the relationships. And it’s where equations are born.

2. Set Match Graphs to 1. [ ]

Drag the cart to the .40 m point. Align one of the draggable horizontal grid lines with the red dot on the position axis.

3. Repeat for the 1.20 m point.

Now consider the time. The two horizontal position lines do double duty in that they pass through the graph for each position twice – once when the cart is traveling to the right and again when it’s traveling through the same points going left.Each of those four intersections were reached at different times. Time is like that. Unlike with space, you can’t double back with time. Or can you. (Cue weird space music.)

4. When is the cart at .40 m? In Figure 8a it appears to be a bit before 1.5 s. It’s a bit hard to read. A vertical grid line would help with this. Create one now. It should intersect the rising section of the graph and the .40-m grid line. This is the point (t1, x1). From your graph find the approximate coordinates, with units, of your point (t1, x1)? Record it below. It won’t be as precise as the computer-generated data table, but we can find data for any point we want. That is, we can interpolate between the data points.

Repeat the previous step for both of the times the cart passes through 1.20-m and then for the second .40-m point. You should now have four vertical grid lines (Figure 8b) and a complete set of data below. Use Zooming to align the grid lines as precisely as possible.

Point 1: t1, x1 =

Point 2: t2, x2 =

Point 3: t3, x3 =

Point 4: t4, x4 =


Figure 8b

5. What’s happening when the graph is at its peak? Where is the cart at this instant?

In the part C, you used the data in your data table to calculate the velocity using Equation 2.

Average or constant velocity, v= Δ xΔt

=x f −xo

t f −t o

Figure 8c shows the same graph with our four numbered points. Note the subscripts.

Figure 8c

6. So how can we use these lines to find the two cart velocities? Consider points 1 and 2. To find the velocity between these two points you’d replace the generic variables, such as xf in Equation 2 with the specific numbered variables, such as x2.

va=Δ xΔ t

=x f −xo

t f−t o =

x2−x1

t 2−t1(3)

Notice the triangles you’ve created. What is x2 – x 1 in terms of one of these triangles? How about t2 – t1?

What property of the graph does (x2 – x1) / (t2 – t1) represent or calculate?


The rising line is straight. What can we say about the velocity between t1 and t2?

The slope of a position vs. time graph at any point equals the velocity at that point (time). If a portion of the line is straight, its slope equals the constant velocity at all times during that time period.

You could also calculate the average velocity between points 1 and 3. What is it? m/s

What is different about this velocity than the others you’ve calculated?

This is still a valid velocity, just as when you calculate the average speed on a road trip. It just doesn’t describe the velocity at any particular point along the travel of the cart. It’s just an overall average.

7. Back to our two constant velocities. You found the t and x data for points 1-4 earlier. Use this data to calculate the constant velocity of the cart while traveling to the right, va, and to the left, vb. Show your calculations below.

va=Δ xΔ t

=¿ x2−x1

t 2−t1 =

1.20 m−.40 m2.7 s−1.3 s

vb=Δ xΔ t

=¿ x4−x3

t 4−t 3 =

.40 m−1.20 m8.9 s−6.3 s

Hopefully that matched with your expectations. And the negative sign indicating the second velocity’s direction comes right from the calculations. These equations behave just like the actual motions. They even give you both magnitude and direction although they aren’t exactly the .60 and -.30 we expected. Graphs can be a bit imprecise as sources of data. But they’re very useful in helping us “see” our data and in interpreting what the data says. This is how a lot of science is done, and not just physical science.

8. What would a graph of velocity vs. time look like? Look at your version of Figure 8c on the apparatus. In between t1 and t2 you had a gradual increase in x from .40 m to 1.20 m. Thus the rising line.

The slope of your graph was constant between t1 and t2, so what would you say about the velocity at various points between those two times? How about between t3 and t4? Write a statement quoting your actual numbers for each case.

Sketch your velocity vs. time graph below for this sequence of motions. A straight edge would be helpful.

Figure 9

9. We’ll use a new tool called “Grapher” to see how you did. It’s used as a companion to many of our pieces of apparatus. You’ll open it from the same place as where you opened the Dynamics Track. It should open in its own window. (Ask your teacher if you have questions. It varies according to the device you’re working with.)Here’s how to get your data from the Dynamics Track to Grapher.


a. In the Dynamics Track [Copy Data to Clipboard] i.e., Click

b. Switch to Grapher. Note the instructions below the data table:

“Click in the box and then Ctrl + v” which means to click and hold down the CTRL (CMD) key and type “v”.

You should now see a very large version of your position vs. time graph.

On the left of the graph, the ordinate, change “axis.” to “Position (m).” Likewise label the abscissa “Time (s).

Title the Graph “Position vs. Time.” Drag it to a good place near the top.

10. Now to create the velocity vs. time graph. This is very difficult. Turn on Graph 2 by clicking the checkbox.Poof! Sorry that was so easy. Grapher senses what you need through contact with your fingertips...

More on that shortly. First label the axes on your new graph “Velocity (m/s)” and “Time (s).” Title it appropriately.

11. Let’s check the numbers. [Examine] (Turn on the “Examine” tool found in the Toolbox.) (Click the check box.)Move your pointer into Graph 1, your (t, x) graph. (Don’t click.) The boxes around Graph 1 and the toolbox (containing Examine for example) become blue. Move your pointer down into Graph 2. Note the color changes. The data in the toolbox refers to a specific graph – the one your pointer is over. This, as well as the other information provided in this activity, is important to know in your future work with Grapher.

Move your pointer back into Graph 1 near the vertical (position) axis. Move it slowly to the right. A little circle called the data tool moves along the graph. In the Examine box, the time, (X), and position, (Y), are displayed as you move your pointer. Notice how it matches your data. It “reads” the graph. (Note the X and Y terms above are generic names for the horizontal and vertical axes. In our case the X corresponds to the time, t, and Y corresponds to the position, x.)

[Tangent] Again move your pointer back into Graph 1. While watching the data tool, move your pointer from the left side to the right side of the graph. The thin slope line attached to the data tool is often hidden in the plotted graph but you can see it where the graph changes direction. What are the values of the slopes of the two line segments that we’ve been studying, and what do they represent? Be sure to include units which come from the two axis units. [Tangent] (You’ll note that the slope readings are not exactly constant. This is because they are being calculated from the data points near the data tool’s position.)

12. Where do these values get their units and where else in Grapher do you see these two values?

If you’ve taken some calculus, one or both of the next two items will be familiar. Don’t panic. A course in calculus is not required for this course. You’ve been learning some of the foundation concepts of it already.

13. So how did Grapher know what we wanted to plot in the second graph?

Graph #1 always plots your Y values vs. your X values. (Position, x vs. time, t in this case).

Graphs #2 and #3 are customizable. They use values calculated from the data plotted in Graph #1. For example, selecting Y^2 for graph 2 will plot the square of each Y-value vs. each X-value. The default is Y´ described below.

[Graphs] (Click the Graphs button.)

In the new box that pops up you should see Y´ selected for graph 2. In calculus terms this means “the first derivative of what’s plotted on the Y-axis (x, position) vs. what’s plotted on the X-axis (t, time).” We know this as “the rate at which position, x, changes with time, t.” That is, Δx/Δt which is just the slope, which in this case is the velocity!

So, since Y´ was the default, Grapher plotted just what you needed, velocity vs. time. Try y^2 which means y2. This just squares each y-value. The other choices are really nifty too! They all might be useful for skate park design.


E. The Area under a Velocity vs. Time Graph of Motion with a Constant VelocityWe just learned that the slope of the (t, x) graph can be used to create the (t, v) graph. Let’s go back in the other direction. Reset the graph 2 setting to Y´.

We’ll continue to use the data from part D, so if it’s gone, please recreate it. Just repeat the initial instructions from part D, and then copy and paste your data to Grapher. Add the second graph so that you have a position vs. time graph and a velocity vs. time graph.

1. [Examine] and [Tangent]. Put your pointer near the vertical (position) axis of Graph #1. When you move your pointer to the right, while still above the short horizontal section, the first variable, the time, increases steadily in the Examine box.The second variable remains fixed. Which quantity does this variable correspond to, time or position? What is the magnitude (amount) of this variable, with units? Why doesn’t it change?

2. Continue to move your pointer horizontally across the graph. The data tool, will take off up the slope. What does this rise in the graph say about the cart’s motion? Reading from the Examine box, what is the maximum value of the cart’s position, with units?

3. What’s the difference (as in subtraction) between the numbers you recorded in #1 and 2? What does it represent?

4. Click, but don’t drag, somewhere in Graph #2. [Integrate] (Welcome to second semester Calculus.) You should see lots of blue. Move your pointer into Graph #2 over near the vertical, velocity axis. Move your pointer slowly about half-way across the graph paying attention to the data tool as it makes its way from one horizontal line above the time axis to the other one below the time axis. What caused this sudden drop? Why do the signs of the Y-values change?

As you near the 10 second point the data tool jumps abruptly upward because of the second bounce. It then moves along a short horizontal segment above the time axis. We aren’t interested in this short segment since we’re just interested in what happens upward until the second bounce. The little “eyeball” in Figure 10 is the last point we’re interested in.

Fig 10

5. Starting with your pointer, and the data tool, at the left side of the graph, at the point where the data tool, (and the graph) first jumps upward, click and drag across the graph until the data tool reaches the “eyeball” but not beyond. See Figure 11. Release. You should now have just two blue rectangles. Record the Area value listed beside Integrate. Ignore a possible negative sign.

Area:

This number needs units? How would that work. If you measure a rectangular table to be 3 ft. by 5 ft., the area will be in ft. × ft. or ft2 – square feet. You get the units from a graph or calculation the same way. You treat the units the same as you treat the numbers. They multiply, divide, etc. Look at the units you provided for each axis and multiply them together and see what you get. Some cancelling of units will occur. Update the area you provided above with the unit remaining after the cancelling.

Fig 11

6. That’s kind of weird. How can two areas add to an area of (approximately) zero?

We’ll take it one rectangle at a time. You want to drag across each one separately to find each of the two areas. For each one you need to start as close as possible to its left edge and release similarly at its right edge. And in the middle, the second rectangle will begin at the point where the first one ended. It won’t be perfect.


Find the area of the first rectangle, with units. Left Area:

7. Does this look familiar? What do you think it represents?

8. What do you predict for the Right area? Right Area Prediction:

Find the second area. Right Area:

Hopefully you see how these two areas can actually add to zero area. These aren’t really areas, but that’s a handy, familiar word for what we see on our graphs. The slope of a graph and the area under a graph are very useful representations of physical quantities. In this case, they represent velocity and displacement respectively. Acceleration and change in velocity will come along soon. Many brilliant minds worked on the development of this mathematics in the 16th through 19th centuries. See http://www.mhhe.com/math/calc/smithminton2e/cd/tools/timeline/ (Click on the images provided for details.)

Table 1 summarizes what you’ve observed so far about these kinematics relationships. We’ll fill out this table as we go.

Graph Slope AreaPosition vs. time, x-t Velocity --------------Velocity vs. time, v-t Coming Soon Displacement, ΔxAcceleration vs. time, a-t -------------- Coming Soon

Table 1 – The Meanings of the Slopes and Areas of x-t, v-t, and a-t graphs

Look back at your graphs and at Table 1 to answer the following.

9. How does a position vs. time graph tell you the magnitude (size) of the velocity? (We usually refer to this as the speed.) How does it tell you the direction of the velocity? Be specific.

10. How does a velocity vs. time graph tell you how far the cart traveled during a given time interval (it’s displacement)? How does it tell you the direction of the motion?

F. Graphical addition of Vectors using Vector ArrowsWe found that we could add the areas under two parts of a v-t graph to find the total displacement of our cart. When it moved with a positive displacement and then a negative one we just added them with signs to find the total displacement. Easy! We can do much the same thing with the vector arrows we used earlier to graphically represent displacement. We use their lengths, drawn to scale, to represent their magnitudes and the directions they point provide their directions.

In the left column of Figure 12 are three displacement vectors, Δx1, Δx2, and Δx3. The first two might represent the motion of our cart moving to the right. The third one might represent its motion after recoiling from the right end of the track. Here’s how we graphically add vectors by drawing scaled vector arrows.

To add vectors graphically connect them together tail to head in any order. Their vector sum, the resultant, is a vector from the tail of the first vector to the head of the last.

1. Here’s an example of adding two vectors pointing in the same directions. Three vectors, Δx1, Δx2, and Δx3 are provided on the left column. Determine the vector sum of displacement vectors Δx1 and Δx2.


http://www.mhhe.com/math/calc/smithminton2e/cd/tools/timeline/

On the right a copy of Δx1 was drawn to scale. Then Δx2 was then drawn starting at the head of Δx1. (We call this adding the vectors “tail to head.” Make sure you understand what this means.) The vector sum, Δx1 + Δx2 or Δx1+2 is a vector from the tail of the first to the head of the last. It’s drawn below the first two vectors to avoid overlapping the vectors. This is fine since its location is immaterial. Its only properties are its magnitude and its direction.

2. This time we want to find Δx1 plus Δx3. These point in opposite directions.

Again, Δx1 is drawn to scale. But when we add Δx3 by starting with its tail at the head of Δx1 we’d be drawing on top of Δx1. So, to avoid that we drop Δx3 down a bit. We then draw the sum, Δx1 + Δx3 as a vector from the tail of Δx1 to the head of Δx3.

Figure 12

You might suggest that this is being made more complicated than it needs to be. However, the situation gets more complicated when we get to vectors in two dimensions. So, getting comfortable with vectors when it seems obvious will make you better prepared when things get less obvious. Vectors are a very big deal in physics.

3. There’s a third possibility. Go back to your answers to questions 6-8 in the previous section. Call the two displacements Δx4 and Δx5.

Using your data from questions 6-8, draw Δx4 and Δx5 below, on the left. Then add the two vectors graphically as was done in Figure 12. State the value for the sum, Δx4 + Δx5 in the box at the right.

Figure 13

4. How is this situation different from the diagrams in Figure 12.?

G. From Data → Graphs → EquationsLet’s do a simple homework type problem based on our dynamics track.

You launch a cart from the left end of track (xo = 7.5 cm) at Vo = +22 cm/s. It maintains this constant speed.

a. How far will it have gone after 6.0 s? b. Where will it be after 6.3 s?


a. You could use motion diagrams for a pretty good estimate. Of course, there are no motion diagrams outside of the lab! Well, Hansel and Gretel did leave a breadcrumb trail... Or if you had a graph? Similar problem. But there is a way that our graphs can provide the answer. From our graphs, we know that for a constant velocity,

v= Δ xΔt

=x f −xo

t f −t o where v is the slope of graph.

a. We have v = 22 cm/s, Δt = 6 s, so

Δx = v Δt = 22 cm/s × 6.0 s = 132 cm (note how the unit cancel as before)

b. Δx = xf – xo = v t where xo = about 7.5 cm (we often drop the Δ and assume to = 0)

xf = xo + v t = 7.5 cm + (22 cm/s × 6.3 s) = 146 cm

So, we can just carry this little equation in our heads and if given the slope (v) we can answer a whole range of questions. Of course, it’s often necessary to do a bit of algebra to solve the basic equation for what we want to know.

Most of the relationships that you’ll work with in this course will fit in one of the seven categories listed in the table below. You’ll find that most equations won’t look just like the ones below because we often combine equations together into new equations for our convenience. And there will also be extra constants added, again for convenience. For example, you’ll soon see the equation x = vo t + ½ a t2. Notice how it combines both a linear term and a quadratic term. You’ll soon see how this equation can be derived from two simpler equations or created straight from a graph.

A Summary of Mathematical Relationships and Related Graphs found in this courseRelation between variables Graph Equation1. No relation Horizontal line y = const.

2. Linear relation with y-intercept Straight, non-horizontal line not through origin

y = mx + bm: slope, b: y-intercept

3. Direct proportion Straight, non-horizontal line through origin

y = mxm: slope

4. Inverse proportion Hyperbola y = k/x

5. Quadratic proportion a. upward opening parabolab. side opening parabola

y = kx2

y2 = kx

6. Inverse square proportion y = k/x2

7. Sinusoidal y = A sin(x) (shown)y = A cos(x)y = A tan(x)


purpose - ket virtual physics labs … · web viewstl-velocity-03-27-key.docx/15print date: ... ...

Documents