Linear Motion or 1D Kinematics

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Nov 13, 2013 (3 years and 7 months ago)

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Linear Motion or 1D
Kinematics

By Sandrine Colson
-
Inam, Ph.D

References:


Conceptual Physics, Paul G. Hewitt, 10
th

edition, Addison
Wesley publisher


http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/1DK
inTOC.html


Outline


The Big Idea


Scalars and Vectors


Distance versus displacement


Speed and Velocity


Acceleration


Describing motion with diagrams


Describing motion with graphs


Free Fall and the acceleration of gravity


Describing motion with equations

The Big Idea


Kinematics

is the science of describing the motion of
objects using words, diagrams, numbers, graphs, and
equations. Kinematics is a branch of mechanics. The goal
of any study of kinematics is to develop sophisticated
mental models which serve to describe (and ultimately,
explain) the motion of real
-
world objects.


Physics is a mathematical science
. The underlying
concepts and principles have a mathematical basis.
Throughout the course of our study of physics, we will
encounter a variety of concepts which have a mathematical
basis associated with them. While our emphasis will often
be upon the conceptual nature of physics, we will give
considerable and persistent attention to its mathematical
aspect.

Scalars and Vectors


Scalars

are quantities which are fully described by a magnitude alone.


Vectors

are quantities which are fully described by both a magnitude and a
direction.


Check Your Understanding:
To test your understanding of this distinction,
consider the following quantities listed below. Categorize each quantity as
being either a vector or a scalar.

QUANTITY

CATEGORY

a. 5 m



b. 30 m/sec, East



c. 5 mi., North



d. 20 degrees Celsius



e. 256 bytes



f. 4000 Calories



Distance versus Displacement


Distance

is a
scalar quantity

which refers to "how much ground an object
has covered" during its motion.


Displacement

is a
vector quantity

which refers to "how far out of place an
object is"; it is the object's overall change in position.


To test your understanding of this distinction, consider the motion depicted
in the diagram below. A physics teacher walks 4 meters East, 2 meters
South, 4 meters West, and finally 2 meters North.



What is the distance covered by the teacher? __________ m


What is his/her displacement? __________ m




Speed versus Velocity


Speed

is a
scalar quantity

which refers to "how fast an object
is moving." Speed can be thought of as the rate at which an
object covers distance. A fast
-
moving object has a high speed
and covers a relatively large distance in a short amount of
time. A slow
-
moving object has a low speed and covers a
relatively small amount of distance in a short amount of time.
An object with no movement at all has a zero speed.


Velocity

is a
vector quantity

which refers to "the rate at which
an object changes its position."

Average Speed versus Instantaneous Speed


Instantaneous Speed

-

speed at any given instant in time.


Average Speed

-

average of all instantaneous speeds; found simply by a
distance/time ratio.


You might think of the instantaneous speed as the speed which the
speedometer reads at any given instant in time and the average speed as the
average of all the speedometer readings during the course of the trip. Since
the task of averaging speedometer readings would be quite complicated (and
maybe even dangerous), the average speed is more commonly calculated as
the distance/time ratio.


Check your understanding
:


While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8
hours. What was her average speed? ________
miles/hr


Now let's consider the motion of
that physics teacher again
. The physics teacher
walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North.
The entire motion lasted for 24 seconds. Determine the average speed and the
average velocity.


Average speed = _______ m/s


Average velocity = _________ m/s in the __________ direction

Acceleration


Acceleration

is a
vector quantity

which is defined as the rate at which an
object changes its
velocity
. An object is accelerating if it is changing its
velocity.




For objects with a constant acceleration, the distance of travel is
directly proportional to the square of the time of travel.



The Direction of the Acceleration Vector


Since acceleration is a
vector quantity
, it has a direction associated with it.
The direction of the acceleration vector depends on two things:


whether the object is speeding up or slowing down


whether the object is moving in the + or
-

direction



The general
RULE OF THUMB

is:


If an object is slowing down, then its acceleration is in the opposite direction of its motion.

Check Your Understanding


To test your understanding of the concept of
acceleration, consider the following problems and the
corresponding solutions. Use the equation for
acceleration to determine the acceleration for the
following two motions.







Acceleration A = _________ m/s/s or m/s
2


Acceleration B = _________ m/s/s or m/s
2

Describing motion with diagrams


Throughout the course, there will be a persistent appeal
to your ability to represent physical concepts in a visual
manner.


The two most commonly used types of diagrams used
to describe the motion of objects are:


ticker tape diagrams


vector diagrams



Ticker Tape Diagrams


A common way of analyzing the motion of objects in physics labs is to perform a
ticker tape
analysis
. A long
tape

is attached to a moving object and threaded through a device that places a
tick upon the tape at regular intervals of time
-

say every 0.10 second. As the object moves, it
drags the tape through the "ticker," thus leaving a trail of dots. The trail of dots provides a history
of the object's motion and therefore a representation of the object's motion.








The distance between dots on a ticker tape represents the object's position change during that
time interval. A large distance between dots indicates that the object was moving fast during that
time interval. A small distance between dots means the object was moving slow during that time
interval. Ticker tapes for a fast
-

and slow
-
moving object are depicted below.


The analysis of a ticker tape diagram will also reveal if the object is moving with a constant
velocity or accelerating. A changing distance between dots indicates a changing velocity and thus
an
acceleration
. A constant distance between dots represents a constant velocity and therefore
no acceleration. Ticker tapes for objects moving with a constant velocity and with an accelerated
motion are shown below.

Check your understanding


Ticker tape diagrams are sometimes referred to as oil drop diagrams. Imagine a
car with a leaky engine that drips oil at a regular rate. As the car travels through
town, it would leave a trace of oil on the street. That trace would reveal
information about the motion of the car. Renatta Oyle owns such a car and it
leaves a signature of Renatta's motion wherever she goes. Analyze the three
traces of Renatta's ventures as shown below. Assume Renatta is traveling from
left to right. Describe Renatta's motion characteristics during each section of the
diagram.

1.



2.



3.









Vector Diagram


Vector diagrams

are diagrams which depict the direction and relative magnitude of a vector
quantity by a vector arrow. Vector diagrams can be used to describe the velocity of a moving
object during its motion. For example, the velocity of a car moving down the road could be
represented by a vector diagram.







In a vector diagram, the magnitude of a vector quantity is represented by the size of the vector
arrow. If the size of the arrow in each consecutive frame of the vector diagram is the same, then
the magnitude of that vector is constant. The diagrams below depict the velocity of a car during
its motion. In the top diagram, the size of the velocity vector is constant, so the diagram is
depicting a motion of constant velocity. In the bottom diagram, the size of the velocity vector is
increasing, so the diagram is depicting a motion with increasing velocity
-

i.e.,
an acceleration
.


Vector diagrams can be used to represent any vector quantity. In future studies, vector diagrams
will be used to represent a variety of physical quantities such as acceleration, force, and
momentum. Be familiar with the concept of using a vector arrow to represent the direction and
relative size of a quantity. It will become a very important representation of an object's motion as
we proceed further in our studies of the physics of motion.


See online animation with varying vector diagrams at
http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/kinema/avd.html


Animation

Describing motion with graphs


Our study of 1
-
dimensional kinematics has been
concerned with the multiple means by which the motion
of objects can be represented. Such means include the
use of words, the use of diagrams, the use of numbers,
the use of equations, and the use of graphs.


The Importance of Slope



The shapes of the position versus time graphs for
these two basic types of motion
-

constant velocity
motion and accelerated motion (i.e., changing velocity)
-

reveal an important principle. The principle is that the
slope of the line on a position
-
time graph reveals useful
information about the velocity of the object. It is often
said, "As the slope goes, so goes the velocity."

Position vs. Time Graphs


The meaning of Shape

See Animations of Various Motions with Accompanying Graphs


Constant Velocity

Positive Velocity


Changing Velocity

Positive Velocity


Constant Velocity

Slow, Rightward (+)


Constant Velocity

Fast, Leftward (+)


Constant Velocity

Slow, Leftward (+)


Constant Velocity

Fast, Rightward (+)


Negative (
-
) Velocity

Slow to Fast


Leftward (
-
) Velocity

Fast to Slow


Check Your Understanding


Use the principle of slope to describe the motion of the objects
depicted by the two plots below. In your description, be sure to
include such information as the direction of the velocity vector
(i.e., positive or negative), whether there is a constant velocity or
an acceleration, and whether the object is moving slow, fast,
from slow to fast or from fast to slow. Be complete in your
description.

Position vs. Time Graphs


The meaning of Slope


The slope of the line on a position versus time
graph is equal to the velocity of the object.





To determine the slope:


Pick two points on the line and determine their coordinates.


Determine the difference in y
-
coordinates of these two points
(
rise
).


Determine the difference in x
-
coordinates for these two points
(
run
).


Divide the difference in y
-
coordinates by the difference in x
-
coordinates (rise/run or slope).


Check Your Understanding:
Determine the velocity (i.e.,
slope) of the object as portrayed by the graph below.



Describing Motion with Velocity vs. Time Graphs
-

Shape


The velocity vs. time graphs for the two types of motion



-

constant velocity and changing velocity (acceleration)



-

can be summarized as follows.









The Importance of Slope



The shapes of the velocity vs. time graphs for these two basic types of motion
-

constant velocity motion and
accelerated motion

(i.e., changing velocity)
-

reveal an
important principle. The principle is that
the slope of the line on a velocity
-
time
graph reveals useful information about the acceleration of the object
. If the
acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration
is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration
is negative, then the slope is negative (i.e., a downward sloping line). This very
principle can be extended to any conceivable motion.

Positive Velocity

Zero Acceleration


Positive Velocity

Positive Acceleration


More about slope

See Animations of Various Motions with Accompanying Graphs


Describing Motion with Velocity vs. Time Graphs
-

Slope

Check Your Understanding


The velocity
-
time graph for a two
-
stage rocket is shown below. Use the
graph and your understanding of slope calculations to determine the
acceleration of the rocket during the listed time intervals.



a. t = 0
-

1 second


b. t = 1
-

4 second


c. t = 4
-

12 second

Determining the Area on a v
-
t Graph


For velocity vs. time graphs, the area bounded by the line and the axes represents the
distance traveled.


The diagram shows three different velocity
-
time graphs; the shaded regions between
the line and the axes represent the distance traveled during the stated time interval.









The method used to find the area under a line on a velocity
-
time graph depends on
whether the section bounded by the line and the axes is a rectangle, a triangle or a
trapezoid. Area formulae for each shape are given below.



The shaded area is representative of the distance traveled by the
object during the time interval from 0 seconds to 6 seconds. This
representation of the distance traveled takes on the shape of a
rectangle whose area can be calculated using the appropriate
equation.

The shaded area is representative of the distance traveled by the
object during the time interval from 0 seconds to 4 seconds. This
representation of the distance traveled takes on the shape of a
triangle whose area can be calculated using the appropriate
equation.

The shaded area is representative of the distance traveled by the
object during the time interval from 2 seconds to 5 seconds. This
representation of the distance traveled takes on the shape of a
trapezoid whose area can be calculated using the appropriate
equation.

Free Fall and the Acceleration of Gravity


A free
-
falling object is an object which is falling under the
sole influence of gravity. Thus, any object which is moving
and being acted upon only by the force of gravity is said to
be "in a state of free fall." This definition of free fall leads to
two important characteristics about a free
-
falling object:


Free
-
falling objects do not encounter air resistance.


All free
-
falling objects (on Earth) accelerate downwards at a rate of
approximately 10 m/s/s (to be exact, 9.8 m/s/s). (acceleration on
Earth of 9.8 m/s/s, downward)


This free
-
fall acceleration can also be demonstrated using
a strobe light and a stream of dripping water. If water
dripping from a medicine dropper is illuminated with a
strobe light and the strobe light is adjusted such that the
stream of water is illuminated at a regular rate


say every
0.2 seconds; instead of seeing a stream of water free
-
falling from the medicine dropper, you will see several
consecutive drops. These drops will not be equally spaced
apart; instead the spacing increases with the time of fall (as
shown in the diagram above), a fact which serves to
illustrate the nature of free
-
fall acceleration.

The Acceleration of Gravity


g = 9.8 m/s/s, downward ( ~ 10 m/s/s, downward)




Thus, velocity changes by 10 m/s every second


If the velocity and time for a free
-
falling object being dropped from a
position of rest were tabulated, then one would note the following
pattern.





Time (s)




Velocity (m/s)




0




0




1




-

9.8




2




-

19.6




3




-

29.4




4




-

39.2




5




-

49.0

Thus


t




v = gt

Representing Free Fall by Graphs

The position vs. time graph for a free
-
falling object is shown below.








Observe that the line on the graph is curved.
A curved line on a position vs. time graph

signifies an accelerated motion. Since a
free
-
falling object is undergoing an acceleration of g = 10 m/s/s (approximate value), you would expect that its position
-
time
graph would be curved. A closer look at the position
-
time graph reveals that the object starts with a small velocity (slow) and
finishes with a large velocity (fast).


A velocity versus time graph for a free
-
falling object is shown below.








Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity versus tim
e g
raph
signifies an accelerated motion. Since a free
-
falling object is undergoing an acceleration (g = 9,8 m/s/s, downward), it would b
e
expected that its velocity
-
time graph would be diagonal. A further look at the velocity
-
time graph reveals that the object start
s with
a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the n
ega
tive
direction and speeding up. An object which is moving in the negative direction and speeding up is said to have a negative
acceleration (if necessary, review the
vector nature of acceleration
). Since the slope of any velocity versus time graph is the
acceleration of the object (
as learned in Lesson 4
), the constant, negative slope indicates a constant, negative acceleration. This
analysis of the slope on the graph is consistent with the motion of a free
-
falling object
-

an object moving with a constant
acceleration of 9.8 m/s/s in the downward direction.

How Fast? and How Far?


Free
-
falling objects are in a state of
acceleration
. Specifically, they are
accelerating at a rate of 10 m/s/s. This is to say that the velocity of a free
-
falling object is changing by 10 m/s every second. If dropped from a
position of rest, the object will be traveling 10 m/s at the end of the first
second, 20 m/s at the end of the second second, 30 m/s at the end of the
third second, etc.

How Fast?


The velocity of a free
-
falling object which has been dropped from a
position of rest is dependent upon the length of time for which it has
fallen. The formula for determining the velocity of a falling object after a
time of t seconds is:





v
f

= g * t



where
g

is the acceleration of gravity (approximately 10 m/s/s on Earth;
its exact value is 9.8 m/s/s). The equation above can be used to calculate
the velocity of the object after a given amount of time.

How Far?


The distance which a free
-
falling object has fallen from a position of rest
is also dependent upon the time of fall. The distance fallen after a time of
t seconds is given by the formula below:





d = 0.5 * g * t
2



where
g

is the acceleration of gravity (approximately 10 m/s/s on Earth;
its exact value is 9.8 m/s/s). The equation above can be used to calculate
the distance traveled by the object after a given amount of time.

The Big Misconception


The acceleration of gravity, g, is the same for all free
-
falling
objects regardless of how long they have been falling, or
whether they were initially dropped from rest or thrown up into
the air.


BUT "Wouldn't an elephant free
-
fall faster than a mouse?"



NO!!


WHY?





All objects free fall at the same rate of acceleration, regardless
of their mass.

Describing Motion with Equations


There are a variety of symbols used in the above equations and each symbol
has a specific meaning.


d



the
displacement

of the object.


t



the
time

for which the object moved.


a



the
acceleration

of the object.


vi



the
initial velocity

of the object.


vf



the
final velocity

of the object.


Each of the four equations appropriately describes the mathematical
relationship between the parameters of an object's motion.

How to use the equations


The process involves the use of a problem
-
solving
strategy which will be used throughout the course. The
strategy involves the following steps:


Construct an informative diagram of the physical situation.


Identify and list the given information in variable form.


Identify and list the unknown information in variable form.


Identify and list the equation which will be used to determine unknown
information from known information.


Substitute known values into the equation and use appropriate
algebraic steps to solve for the unknown information.


Check your answer to insure that it is reasonable and mathematically
correct.

Example A


Ima Hurryin is approaching a stoplight moving with a velocity of +30.0 m/s. The light
turns yellow, and Ima applies the brakes and skids to a stop. If Ima's acceleration is
-
8.00 m/s2, then determine the displacement of the car during the skidding process.
(Note that the direction of the velocity and the acceleration vectors are denoted by a +
and a
-

sign.)

Solution for A


The solution to this problem begins by the construction of an informative diagram
of the physical situation. This is shown below. The second step involves the
identification and listing of known information in variable form. Note that the vf
value can be inferred to be 0 m/s since Ima's car comes to a stop. The initial
velocity (vi) of the car is +30.0 m/s since this is the velocity at the beginning of the
motion (the skidding motion). And the acceleration (a) of the car is given as
-

8.00
m/s2. (Always pay careful attention to the + and
-

signs for the given quantities.)
The next step of the
strategy

involves the listing of the unknown (or desired)
information in variable form. In this case, the problem requests information about
the displacement of the car. So d is the unknown quantity. The results of the first
three steps are shown in the table below.

Diagram:





Given:

v
i

= +30.0 m/s

v
f

= 0 m/s

a =
-

8.00 m/s
2

Find:

d = ??

Solution for A
-

end


The next step of the
strategy

involves identifying a kinematic equation which would allow you to determine the
unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the
equation which contains the three known and the one unknown variable. In this specific case, the three known
variables and the one unknown variable are vf, vi, a, and d. Thus, you will look for an equation which has these
four variables listed in it. An inspection of the
four equations above

reveals that the equation on the top right
contains all four variables.





Once the equation is identified and written down, the next step of the
strategy

involves substituting known
values into the equation and using proper algebraic steps to solve for the unknown information. This step is
shown below.


(0 m/s)2 = (30.0 m/s)2 + 2*(
-
8.00 m/s2)*d


0 m2/s2 = 900 m2/s2 + (
-
16.0 m/s2)*d


(16.0 m/s2)*d = 900 m2/s2
-

0 m2/s2


(16.0 m/s2)*d = 900 m2/s2


d = (900 m2/s2)/ (16.0 m/s2)


d = (900 m2/s2)/ (16.0 m/s2)


d = 56.3 m


The solution above reveals that the car will skid a distance of 56.3 meters. (Note that this value is rounded to
the third digit.)


The last step of the
problem
-
solving strategy

involves checking the answer to assure that it is both reasonable
and accurate. The value seems reasonable enough. It takes a car a considerable distance to skid from 30.0 m/s
(approximately 65 mi/hr) to a stop. The calculated distance is approximately one
-
half a football field, making this
a very reasonable skidding distance. Checking for accuracy involves substituting the calculated value back into
the equation for displacement and insuring that the left side of the equation is equal to the right side of the
equation. Indeed it is!

More Practice Problems at
http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L6d.html


SUMMARY: See
Hand
-
outs