representing motion represent motion through the use of words, motion diagrams, and graphs. use the...
TRANSCRIPT
Representing Motion
Represent motion through the use of words, motion diagrams, and graphs.
Use the terms position, distance, displacement, and time interval in a scientific manner to describe motion.
Chapter
2
In this chapter you will:
Table of Contents
Chapter 2: Representing Motion
Section 2.2: Where and When?
Section 2.3: Position-Time Graphs
Section 2.4: How Fast?
Assignments
• Read Chapter 2.
• Study Guide Chapter 2 & 3 due the day before the test.
Chapter
2
Picturing Motion
Open your text to page 31. Read the Launch Lab.
Your teacher will give you your next instructions.
Launch Lab: Which Car is Faster?
Section
2.1
Where and When?
Define coordinate systems for motion problems.
Recognize that the chosen coordinate system affects the sign of objects’ positions.
Define displacement.
Determine a time interval.
Use a motion diagram to answer questions about an object’s position or displacement.
In this section you will:
Section
2.2
Where and When?
A coordinate system tells you the location of the zero point of the variable you are studying and the direction in which the values of the variable increase.
Coordinate Systems
Section
2.2
The origin is the point at which both variables have the value zero.
Where and When?
Observe the motion diagram of a runner. The origin, represented by the zero end of the measuring tape, could be placed 5 m to the left of the tree.
The motion is in a straight line, thus, your measuring tape should lie along that straight line. The straight line is an axis of the coordinate system.
Coordinate Systems
Section
2.2
Where and When?
You can indicate how far away an object is from the origin at a particular time on the simplified motion diagram by drawing an arrow from the origin to the point representing the object, as shown in the figure.
The arrow shown in the figure represents the runner’s position, which is the separation between an object and the origin.
Coordinate Systems
Section
2.2
The length of how far an object is from the origin indicates its distance from the origin.
Where and When?
Coordinate Systems
Section
2.2
The arrow points from the origin to the location of the moving object at a particular time.
A position 9 m to the left of the tree, 5 m left of the origin, would be a negative position, as shown in the figure below.
Where and When?
Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows.
Vectors and Scalars
Section
2.2
Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars.
To add vectors graphically, the length of a vector should be proportional to the magnitude of the quantity being represented. So it is important to decide on the scale of your drawings.
The important thing is to choose a scale that produces a diagram of reasonable size with a vector that is about 5 –10 cm long.
Where and When?
The vector that represents the sum of the other two vectors is called the resultant.
Vectors and Scalars
Section
2.2
The resultant always points from the tail of the first vector to the tip of the last vector.
Where and When?
The difference between the initial and the final times is called the time interval.
Time Intervals and Displacement
Section
2.2
The common symbol for a time interval is ∆t, where the Greek letter delta, ∆, is used to represent a change in a quantity.
Where and When?
The time interval is defined mathematically as:
Time Intervals and Displacement
Section
2.2
Although subscripts i and f are used to represent the initial and final times, they can be initial and final times of any time interval you choose.
Also of importance is how the position changes. The symbol d may be used to represent position.
In physics, a position is a vector with its tail at the origin of a coordinate system and its tip at the place where the object is located at that time.
Where and When?
The figure below shows ∆d, an arrow drawn from the runner’s position at the tree to his position at the lamppost.
Time Intervals and Displacement
Section
2.2
The change in position during the time interval between ti and tf
is called displacement.
Where and When?
The length of the arrow represents the distance the runner moved, while the direction the arrow points indicates the direction of the displacement.
Displacement is mathematically defined as follows:
Time Intervals and Displacement
Section
2.2
Displacement is equal to the final position minus the initial position.
Question: How do you subtract vectors?
Where and When?
To subtract vectors, reverse the subtracted vector and then add the two vectors. This is because A – B = A + (–B).
The figure a below shows two vectors, A, 4 cm long pointing east, and B, 1 cm long also pointing east. Figure b shows –B, which is 1 cm long pointing west. The resultant of A and –B is 3 cm long pointing east.
Section
2.2
Where and When?
Distance is a scalar quantity; it does not describe direction. Distance is simply how far an object moved.
Example: The odometer on your car tells you the distance traveled.
Displacement is a vector ; it has magnitude and direction. It is calculated by subtracting the final minus initial position.
Example: If you walk 50 meters, and end up where you start, your displacement is zero, but your distance traveled is 50 m.
The displacement vector is always drawn with its flat end, or tail, at the earlier position, and its point, or tip, at the later position.
The Difference Between Distance and Displacement
Section
2.2
Section Check
Differentiate between scalar and vector quantities?
Question 1
Section
2.2
Section Check
Answer 1
Section
2.2
Quantities that have both magnitude and direction are called vectors, and can be represented by arrows. Quantities that are just numbers without any direction, such as time, are called scalars.
Section Check
What is displacement?
Question 2
Section
2.2
A. The vector drawn from the initial position to the final position of the motion in a coordinate system.
B. The length of the distance between the initial position and the final position of the motion in a coordinate system.
C. The amount by which the object is displaced from the initial position.
D. The amount by which the object moved from the initial position.
Section Check
Answer: A
Answer 2
Section
2.2
Reason: Options B, C, and D are all defining the distance of the motion and not the displacement. Displacement is a vector drawn from the starting position to the final position.
Refer the figure and calculate the time taken by the car to travel from one signal to another signal?
• Insert the figure shown for question 4.
Question 3
Section
2.2 Section Check
A. 20 min
B. 45 min
C. 25 min
D. 5 min
Section Check
Answer: C
Answer 3
Section
2.2
Reason: Time interval t = tf - ti
Here tf = 01:45 and ti = 01:20
Therefore, t = 25 min
Position-Time Graphs
Section 2.3
Position vs. Time Graphs
Section
2.3
Position-Time Graphs
Develop position-time graphs for moving objects.
Use a position-time graph to interpret an object’s position or displacement.
Make motion diagrams, pictorial representations, and position-time graphs that are equivalent representations describing an object’s motion.
In this section you will:
Section
2.3
Position-Time Graphs
Position Time Graphs
Section
2.3
Click image to view movie.
Graphs of an object’s position and time contain useful information about an object’s position at various times and can be helpful in determining the displacement of an object during various time intervals.
Position-Time Graphs
Using a Graph to Find Out Where and When
Section
2.3
The data in the table can be presented by plotting the time data on a horizontal axis and the position data on a vertical axis, which is called a position-time graph.
To draw the graph, plot the object’s recorded positions. Then, draw a line that best fits the recorded points. This line represents the most likely positions of the runner at the times between the recorded data points.
Position-Time Graphs
Using a Graph to Find Out Where and When
Section
2.3
The symbol d represents the instantaneous position of the object—the position at a particular instant.
Position-Time Graphs
Words, pictorial representations, motion diagrams, data tables, and position-time graphs are all representations that are equivalent. They all contain the same information about an object’s motion.
Depending on what you want to find out about an object’s motion, some of the representations will be more useful than others.
Equivalent Representations
Section
2.3
Position-Time Graphs
Considering the Motion of Multiple Objects
• In the graph, when and where does runner B pass runner A?
Section
2.3
• In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B.
• These lines intersect at 45.0 s and at about 190 m.
Solution: B passes A about 190 m beyond the origin, 45.0 s after A has passed the origin.
Position-Time GraphsSection
2.3
Considering the Motion of Multiple Objects
A position-time graph of an athlete winning the 100-m run is shown. Estimate the time taken by the athlete to reach 65 m.
Question 1
Section
2.3 Section Check
A. 6.0 s
B. 6.5 s
C. 5.5 s
D. 7.0 s
Section Check
Answer: B
Answer 1
Section
2.3
Reason:
Draw a horizontal line from the position of 65 m to the line of best fit. Draw a vertical line to touch the time axis from the point of intersection of the horizontal line and line of best fit. Note the time where the vertical line crosses the time axis. This is the estimated time taken by the athlete to reach 65 m.
A position-time graph of an athlete winning the 100-m run is shown. What was the instantaneous position of the athlete at 2.5 s?
Question 2
Section
2.3 Section Check
A. 15 m
B. 20 m
C. 25 m
D. 30 m
Section Check
Answer: C
Answer 2
Section
2.3
Reason:
Draw a vertical line from the position of 2.5 m to the line of best fit. Draw a horizontal line to touch the position axis from the point of intersection of the vertical line and line of best fit. Note the position where the horizontal line crosses the position axis. This is the instantaneous position of the athlete at 2.5 s.
From the following position-time graph of two brothers running a 100-m run, analyze at what time do both brothers have the same position. The smaller brother started the race from the 20-m mark.
Question 3
Section
2.3 Section Check
Section Check
Answer 3
Section
2.3
The two brothers meet at 6 s. In the figure, we find the intersection of line representing the motion of one brother with the line representing the motion of other brother. These lines intersect at 6 s and at 60 m.
How Fast?
Define velocity.
Differentiate between speed and velocity.
Create pictorial, physical, and mathematical models of motion problems.
In this section you will:
Section
2.4
How Fast?
Suppose you recorded two joggers on one motion diagram, as shown in the below figure. From one frame to the next, you can see that the position of the jogger in red shorts changes more than that of the one wearing blue.
Velocity
Section
2.4
In other words, for a fixed time interval, the displacement, ∆d, is greater for the jogger in red because she is moving faster.
She covers a larger distance than the jogger in blue does in the same amount of time.
Now, suppose that each jogger travels 100 m. The time interval, ∆t, would be smaller for the jogger in red than for the one in blue.
How Fast?
Velocity
Section
2.4
How Fast?
Recall from Chapter 1 that to find the slope, you first choose two points on the line.
Next, you subtract the vertical coordinate (d in this case) of the first point from the vertical coordinate of the second point to obtain the rise of the line.
After that, you subtract the horizontal coordinate (t in this case) of the first point from the horizontal coordinate of the second point to obtain the run.
Finally, you divide the rise by the run to obtain the slope.
Average Velocity
Section
2.4
How Fast?
The slopes of the two lines are found as follows:
The unit of the slope is meter per second. In other words, the slope tells how many meters the runner moved in 1 s.
Average Velocity
Section
2.4
d dt t
f i
f i
Red slope =
6.0 m 2.0 m3.0 s 1.0 s
=
= 2.0 m/s
d dt t
f i
f i
Blue slope =
3.0 m 2.0 m3.0 s 2.0 s
=
= 1.0 m/s
How Fast?
The slope of a position-time graph for an object is the object’s average velocity and is represented by the ratio of the change of position to the time interval during which the change occurred.
Average Velocity
Section
2.4
Average velocity is defined as the change in position, divided by the time during which the change occurred.
The symbol ≡ means that the left-hand side of the equation is defined by the right-hand side.
How Fast?
The slope of the position-time graph on the right is –5.0 m/s. It indicates the average velocity of the object and not its speed.
The object moves in the negative direction at a rate of 5.0 m/s.
Average Velocity
Section
2.4
It is a common misconception to say that the slope of a position-time graph gives the speed of the object.
How Fast?
The absolute value of the slope of a position-time graph tells you the average speed of the object, that is, how fast the object is moving.
Average Speed
Section
2.4
vv
The sign of the slope tells you in what direction the object is moving. The combination of an object’s average speed, , and the direction in which it is moving is the average velocity .
If an object moves in the negative direction, then its displacement is negative. The object’s velocity will always have the same sign as the object’s displacement.
How Fast?
Average Speed: Example 1The graph describes the motion of a student riding his skateboard along a smooth, pedestrian-free sidewalk. What is his average velocity? What is his average speed?
Section
2.4
Use the Problem Solving StrategyGUESS
How Fast?
Average Speed: Example 1
Section
2.4
Average Speed: Example 1
Given: Study the graph. Look at the labels on the axes. Identify the coordinate system.
How Fast?Section
2.4
Unknown: Identify the unknown variables.
Unknowns:
Equation: Find the average velocity using two points on the line.
How Fast?Section
2.4
Use magnitudes with signs indicating directions.
Decide which two points on the graph you will use.
Average Speed: Example 1
More givens, as read off the graph: d2 = 12.0 m, d1 = 6.0 m, t2 = 7.0 s, t1 = 3.5 s: Substitute & Solve:
v = 12.0 m – 6.0 m 7.0 s – 3.5 s
= 1.7 m/s
How Fast?Section
2.4
Average Speed: Example 1
The velocity is 1.7 m/s.The speed is 1.7 m/s.
How Fast?
Are the units correct?
m/s are the units for both velocity and speed.
Do the signs make sense?
The positive sign for the velocity agrees with the coordinate system. No direction is associated with speed.
Average Speed: Example 1
Sense: Does the answer make sense?
Section
2.4
How Fast?
A motion diagram shows the position of a moving object at the beginning and end of a time interval. During that time interval, the speed of the object could have remained the same, increased, or decreased. All that can be determined from the motion diagram is the average velocity.
The speed and direction of an object at a particular instant is called the instantaneous velocity.
Section
2.4
Instantaneous Velocity
The term velocity refers to instantaneous velocity and is represented by the symbol v.
How Fast?
On a position-time graph, you can find the average velocity by:
- plotting the position for t1 and t2
- drawing a straight line between these points
- calculating the slope of the line
Average Velocity
Section
2.4
po
siti
on
(m
)
time (s) t1 t2
v = slope
How Fast?
On a position-time graph, you can find the instantaneous velocity by:
- plotting the position for t
- drawing a tangent line at that point
- calculating the slope of the line
Instantaneous Velocity
Section
2.4
po
siti
on
(m
)
time (s) t
v = slope
Any time you graph a straight line, you can find an equation to describe it.
How Fast?
Using Equations
Section
2.4
For a position-time graph, the equation:
y = mx + b becomes
d = vt + di
How Fast?
An object’s position is equal to the average velocity multiplied by time plus the initial position.
Equation of Motion for Average Velocity
Using Equations
Section
2.4
This equation gives you another way to represent the motion of an object.
Note that once a coordinate system is chosen, the direction of d is specified by positive and negative values, and the boldface notation can be dispensed with, as in “d-axis.”
Section Check
Which of the following statement defines the velocity of the object’s motion? (Hint: which is the vector?)
Question 1
Section
2.4
A. The ratio of the distance covered by an object to the respective time interval.
B. The rate at which distance is covered.
C. The distance moved by a moving body in unit time.
D. The ratio of the displacement of an object to the respective time interval.
Section Check
Answer: D
Answer 1
Section
2.4
Reason: Options A, B, and C define the speed of the object’s motion. Velocity of a moving object is defined as the ratio of the displacement (d) to the time interval (t).
Section Check
Which of the statements given below is correct?
Question 2
Section
2.4
A. Average velocity cannot have a negative value.
B. Average velocity is a scalar quantity.
C. Average velocity is a vector quantity.
D. Average velocity is the absolute value of the slope of a position-time graph.
Section Check
Answer: C
Answer 2
Section
2.4
Reason: Average velocity is a vector quantity, whereas all other statements are true for scalar quantities.
The position-time graph of a car moving on a street is as given here. What is the average velocity of the car?
Question 3
Section
2.4 Section Check
A. 2.5 m/s
B. 5 m/s
C. 2 m/s
D. 10 m/s
Section Check
Answer: C
Answer 3
Section
2.4
Reason: Average velocity of an object is the slope of the position-time graph.