constant velocity particle model packet

Name: Date: Physics Mrs. Segal

Constant Velocity Particle Model

The front of each model packet should serve as a storehouse for things you’ll want to be able to quickly look up later.

Physical Quantity

Description Symbol Units

©Modeling Instruction 2010 1 U2 Constant Velocity


Constant Velocity Particle Model Lab: Motorized Car Experiment

What could we measure? How could we measure it?

Take data in an organized labeled way. 1. Note which quantities you kept constant and what the values of each of them are

2. Make a table of the two variables you are looking to find the relationship between.


Sketch and label the experiment setup:

The Objective: To determine the relationship between


Constant Velocity Particle Model Lab: Motorized Car ExperimentGraph your data to see if there is a relationship

If it is linear, find the… equation of best fit line: _________________________________________________

Be sure to: Use pencil Label your axes with symbols and units Give the graph a title (“[vertical axis variable] vs. [horizontal axis variable]”) Draw a best fit line (don’t connect the dots). Find the slope using points on the line (not data points). Write the equation of the line using the variables from your axes (don’t default to “y and

x”); make sure the slope and intercept have the correct units attached to the numbers. Always put units on numbers, but never on variables.



Constant Velocity Particle Model Lab: Motorized Car ExperimentNote taking Guide for Whiteboard Meetings

Pre-

Mee

ting

Our group’s conclusionsMathematical model:

The slope represents:

The y-intercept represents:

Interesting things we noticed:

Problems we had:

I’m wondering about…Are the results what you were expecting?

I noti

ce th

at …

These things are the sameon each group’s whiteboard

These things are differenton each group’s whiteboard

Refle

ction

s

Consensus notes on our class discussion I’m still confused about…



Constant Velocity Particle Model Reading: Motion Maps

A motion map represents the position, velocity, and acceleration of an object at various clock readings. (At this stage of the class, you will be representing position and velocity only.)

Suppose that you took a stroboscopic picture of a car moving to the right at constant velocity where each image revealed the position of the car at one-second intervals.

This is the motion map that represents the car. We model the position of the object with a small point. At each position, the object's velocity is represented by a vector.

If the car were traveling at greater velocity, the strobe photo might look like this:

The corresponding motion map has the points spaced farther apart, and the velocity vectors are longer, implying that the car is moving faster.

If the car were moving to the left at constant velocity, the photo and motion map might look like this:

More complicated motion can be represented as well.

Here, an object moves to the right at constant velocity, stops and remains in place for two seconds, then moves to the left at a slower constant velocity.



Constant Velocity Particle Model Reading: Motion Maps

Consider the interpretation of the motion map below. At time t = 0, cyclist A starts moving to the right at constant velocity, at some position to the right of the origin.

Cyclist B starts at the origin and travels to the right at a constant, though greater velocity. At t = 3 s, B overtakes A (i.e., both have the same position, but B is moving faster).

A graphical representation of the behavior of cyclists A and B would like this:

Throughout this semester, you will be representing the behavior of objects in motion in multiple ways: diagrammatically (motion maps), graphically and algebraically.

Hints for drawing your own motion maps:

1. Draw dots indicating the position of the object at equal time intervals, i.e. each second.

2. Attach arrows to the dots indicating the direction of motion. Make the arrow length half of the space between the dots to make your motion map easy to read.

3. When an object is stopped for several time intervals, draw multiple dots at the same position.

4. Make sure your sequence of arrows has a logical flow so that the motion is clearly communicated.



Constant Velocity Particle Model Practice 1: Motion Maps and Position vs. Time Graphs

1. Given the following position vs. time graph, draw a motion map with one dot for each second.

Describe the motion of the object in words:

2. Given the following motion map, where positions have been recorded with one dot each second, draw a position vs. time graph.


0 m 5 m

+

Position vs. Time

time (seconds)



Describe the motion of the object in words:




3. Consider the position vs. time graph below for cyclists A and B.

.

A

B

5 t (s)

x (m

)

a. Do the cyclists start at the same point? How do you know? If not, which is ahead?

b. At t = 7s, which cyclist is ahead? How do you know?

c. Which cyclist is traveling faster at 3s? How do you know?

d. Are their velocities equal at any time? How do you know?

e. What is happening at the intersection of lines A and B?

f. Draw a motion map for cyclists A and B. 0 m

+




4. Consider the new position vs. time graph below for cyclists A and B.

a. How does the motion of the cyclist A in this graph compare to that of A in question 3?

b. How does the motion of cyclist B in this graph compare to that of B in question 3?

c. Which cyclist has the greater speed? How do you know?

d. Describe what is happening at the intersection of lines A and B.

e. Which cyclist has traveled further during the first 5 seconds? How do you know?

f. Draw a motion map for cyclists A and B.

0 m

+


x (m)

t (s)

A

25

x (m)

t (s)

C

10

10

x (m)

t (s)

B

5

x (m)

t (s)

D

10

3

10

x (m)

t (s)

F

10

5

3 10

x (m)

t (s)

E

15

10

0 0

0 0

0 0

0 0

0 0

0 0

1010



5. Please rank the following, make sure you know the difference between displacement and odometer reading.

a. Rank these situations from greatest to least based on which shows the greatest displacement during the time from 0 to 10 seconds. Use the > and = signs, but do not use the < sign.

Briefly explain your reasoning for your ranking:

b. Rank these situations from greatest to least based on which shows the greatest distance traveled (odometer reading) during the time from 0 to 10 seconds. Use the > and = signs, but do not use the < sign.

Briefly explain your reasoning for your ranking:



Constant Velocity Particle Model Ultrasonic Motion Detector Lab:Multiple Representations of Motion

For each of the following situations:1. Fill out each of the 3 missing boxes (velocity vs. time graph, written description, and motion map)

based on the information in the position vs. time graph. DO THIS FIRST, BEFORE YOU USE THE MOTION SENSOR! The written description should include: starting position, direction moved, type of motion, and velocity.

2. Move, relative to the motion detector, so that you produce a position vs. time graph that closely approximates the graph shown.

3. Check to see that the information in each of the 3 boxes is consistent with the way you moved and the graphs shown on the computer. Using a different colored pen or pencil, correct your predictions if necessary.

1. Written Description:

Motion Map:


Motion Map:





Motion Map:


Motion Map:




For each of the following situations1. Fill out each of the 3 missing boxes (position vs. time graph, written description, and motion map)

based on the information in the position vs. time graph2. Move, relative to the motion detector, so that you produce a velocity vs. time graph that closely

approximates the graph shown. 3. Check to see that the information in each of the 3 boxes is consistent with the way you moved and the

graphs shown on the computer. Make any adjustments as necessary


Motion Map:


Motion Map:





Motion Map:


Motion Map:





Motion Map:



Constant Velocity Particle Model Practice 2: Motion Maps and Velocity vs. Time Graphs

Sketch velocity vs. time graphs and motion maps corresponding to the following descriptions of the motion of an object.

1. The object is moving in the positive direction at a constant (steady) speed.

Motion Map:

0 m

+

time

2. The object is standing still.

Motion Map:

0 m

+time

3. The object moves in the negative direction at a steady speed for 10s, then stands still for 10s.

Motion Map:

0 m

+ time

4. The object moves in the positive direction at a steady speed for 10s, reverses direction and moves back toward the negative direction at the same speed.

Motion Map:

0 m

+

time



Constant Velocity Particle Model Practice 3: Position vs. Time Graphs and Velocity vs. Time Graphs

Draw the velocity vs. time graphs for an object whose motion produced the position vs. time graphs shown below at left.

1.

Distance (with units):Displacement (with units):Slope(s) (with units):

Velocity/Velocities (with units):Area between line and time axis (with units):

2.





Constant Velocity Particle Model Practice 3: Position vs. Time Graphs and Velocity vs. Time Graphs

3.



4. For many graphs, both the slope of the line and the area between the line and the horizontal axis have physical meanings.

a. Calculate the slope of each position vs. time graph. What does the slope of a position time graph tell you about the motion of an object?

b. Looking at the velocity time graphs, determine the units for a square of area on the graph.

c. Calculate the area between each velocity graph and the horizontal axis. What does the area under the velocity-time graph tell you about the motion of an object?



Constant Velocity Particle Model Practice 4:Position vs. Time Graphs and Average Speed/Velocity

1. Robin, roller skating down a marked sidewalk, was observed at the following positions at the times listed below:

a. Plot a position vs. time graph for the skater.

b. Explain how you can use the graph to determine how far she was from the origin at t = 6s.

c. Write a mathematical model that describes the skater's motion.

d. Was her speed constant over the entire interval? How do you know?


t (s)

t (s) x (m)

0.0 10.0

1.0 12.0

2.0 14.0

5.0 20.0

8.0 26.0

10.0 30.0



2. In a second trial, the timer started her watch a bit sooner. The following data were obtained:

a. Plot the position vs. time graph for the skater.

b. How far from the origin was the skater at t = 5s? How do you know?

c. Was the skater’s speed constant? If so, what was it?

d. Your friend tells you that Robin was moving faster during the second trial because her position at a clock reading of 8 s is greater in the second trial than the first. Is she right?


t (s)

t (s) x (m)

0.0 4.0

2.0 10.0

4.0 16.0

6.0 22.0

8.0 28.0

10.0 34.0



3. Suppose now that our skater was observed in a third trial. The following data were obtained:

a. Plot the position vs. time graph for the skater.

b. What do you think is happening during the time interval: t = 4s to t = 6s? How can you support your idea?

c. What do you think is happening during the time interval: t = 6s to t = 10s? How can you support your idea?

d. Determine the skater's average velocity from t = 0s to t = 16s. (Average velocity is the displacement (final position minus initial position) divided by time elapsed.)

e. Determine the skater's average speed from t = 0s to t = 16s. (Average speed is the distance traveled along the path (change in odometer reading) divided by time elapsed.)


t (s)

t (s) x (m)0.0 0.02.0 2.04.0 4.06.0 4.08.0 3.0

10.0 2.012.0 2.014.0 5.016.0 8.0



f. In what situation is average speed a better measure of motion than average velocity?

g. In what situation is average velocity a better measure of motion than average speed?


x (m)

t (s)

A

25

x (m)

t (s)

C

10

10

x (m)

t (s)

B

5

x (m)

t (s)

D

10

3

10

x (m)

t (s)

F

10

5

3 10

x (m)

t (s)

E

15

10

0 0

0 0

0 0

0 0

0 0

0 0

1010



4. Consider the following position vs. time graphs

a. Rank the graphs according to which show the greatest average velocity from the beginning to the end of the motion. (Ties are possible.)

Greatest v 1________ 2________ 3________ 4________ 5________ 6________ Least v

Explain your reasoning for your ranking:

b. Rank the graphs according to which show the greatest average speed from the beginning to the end of the motion.

Greatest 1________ 2________ 3________ 4________ 5________ 6________ Least

Explain your reasoning for your ranking:


x

t

v

t


Constant Velocity Particle Model Practice 5: Pulling it Together

1. This motion map shows the position of an object once every second. From the motion map, answer the following:

a. Describe the motion of the object in words.

b. Represent the motion with a quantitative x vs. t graph.

c. Represent the motion with a quantitative v vs. t graph.

d. Write a mathematical expression that represents the relationship between position and time.

e. Write a mathematical expression that represents the relationship between velocity and time.

f. Cross hatch the area under the velocity-time graph. What are the units of this area? Describe what the area under the v-t graph represents and find its value.



Constant Velocity Particle Model Practice 5: Pulling it Together

2. From the position vs. time data below, answer the following questions.

t(s)

x(m)

0 01 22 43 44 75 106 107 108 59 0

a. Construct a graph of position vs. time

b. Construct a graph of velocity vs. time.

c. . Draw a motion map for the object.

d. Determine the displacement from t = 3.0s to 5.0s using the velocity vs. time graph.

e. Determine the displacement from t = 7.0 s to 9.0 s using the velocity vs. time graph.

f. Determine the average velocity from t = 4 s to 8 s.

g. Determine the average speed from t = 4 s to 8 s.


t

t

x

vMotion map:

Written description:

t

t

x

vMotion map:



Constant Velocity Particle Model Practice 6: Multiple Representations of Motion

Given one motion representation, supply the missing motion representations.


1.

2.

t

t (s)

x

v (m/s) Motion map:


4.

2 4 8

+2

-1

t (s)

t

x

vMotion map:

Written description:3.

3 6 8




5.

t

t

x

vMotion map:


Object moves with constant positive velocity for 4 seconds. Then, it stops for 2 seconds and returns to the initial position in 2 seconds.

t

t

x

vMotion map: ∆t = 1 s


0m 1m




t

t

x

vMotion map: ∆t = 1s


A

B

3m 4m

t

t

x

vMotion map:





7.


Constant Velocity Particle Model Practice 7: Applying the Model

1. Read the following three situations and consider if the Constant Velocity Particle Model (CVPM) applies.

I. A Mac Truck starts from rest and reaches a speed of 8.0 m/s in 20 seconds.II. A dune buggy travels for 20 seconds at a speed of 8.0 m/s.

III. A driver sees a deer in the road ahead and applies the brakes. The car slows to a stop from 8.0 m/s in 20 seconds.

a. For each of the three above problems, say whether CVPM applies and explain your reasoning.

b. Choose one of the problems for which CVPM applies. For the problem you selected, draw at least three diagrams and/or graphs to illustrate the situation. Choose the diagrams and graphs that you find most useful.

Using the constant velocity particle model, solve for any unknown quantities. Show your work and use units.



Constant Velocity Particle Model Practice 7: Applying the Model

2. The graph below shows the velocity vs. time graph for a toy dune buggy which started 20 cm from the edge of its track. Assume that edge of the track is the origin.

a. Determine the change in position from t = 2 sec to 3.5 sec. Clearly indicate how the change in position shows up on the velocity graph. Show your work and use units!

b. Determine the change in position from t = 5 sec to 6 sec. Clearly indicate how the change in position shows up on the velocity graph. Show your work and use units!

c. Construct a quantitative position-time graph for the motion. Assume a position of 20 cm at t = 0. Be sure to accurately number the scale on the position axis.

d. Draw a motion map for this motion. On your motion map, clearly indicate the displacements determined in parts (a) and (b).



Constant Velocity Particle Model Lab Practicum: Dueling Cars

Find a group that used a different speed car than you used. Get together and see if you can figure out how to predict where the cars will crash when separated a certain distance.

When you are ready with a method, I will take the cars and tell you the distance.

Come up with a prediction and we’ll test it!

Solve this using at least two different methods



Constant Velocity Particle Model: Review Sheet

1. Consider the following position vs. time graph.a. Determine the velocity of the object.

b. Write a mathematical model to describe the motion of the object.

2. Shown below is a velocity vs. time graph for an object.

a. a. Describe the motion of the object.

b. Draw a corresponding position vs. time graph. Choose an appropriate scale for the axes. You may assume the object starts from the origin.

c. How far did the object travel in the interval t =1s to t =2s?

d. Find the displacement from t = 0s to t = 5 s. Explain how you got your answer.

e. Find the average velocity from t = 0s to t = 5 s. Explain how you got your answer.

f. Find the average speed from t = 0s to t = 5 s. Explain how you got your answer.




3. A bird travels toward the origin, then suddenly reverses direction.

a. Find the velocity from t = 30s to t = 40s.

b. Find the velocity from t = 40s to t = 50s.

c. Determine the average speed from t = 30s to t = 50s.

d. Determine the average velocity from t = 30s to t = 50s.

e. Find the velocity at t = 35 seconds.




4. A basketball initially travels at 3 meters per second for 3 seconds:

a. Describe the motion of the ball after t = 3 seconds.

b. Draw a quantitative motion map that models the motion of the object.

0 m

+

c. How far did the ball travel from t = 3s to t = 7s?

5. A racecar reaches a speed of 90 m/s after it is 450 meters past the starting line. If the car travels at a constant speed of 90 m/s for the next 12.5 s, how far will the car be from the starting line? Use the appropriate mathematical model and show how units cancel.



Constant Velocity Particle Model: Model Summary


constant velocity particle model packet

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