Topic #3. Motion in a Straight Line
1. Solving Physics Problems {using a problem-solving format like GFESA}
2. Motion
3. Scalar and Vector Quantities
4. Vector Addition - Graphical Method
5. Distance and Displacement
6. Instantaneous Speed and Velocity
7. Average Speed and Velocity
8. Acceleration
9. Final Velocity After Uniform Acceleration
10. Displacement During Uniform Acceleration
11. Acceleration Due to Gravity
Notes should include:
Solving Physics Problems: Solving a problem requires critical reading skills and critical
problem solving skills. Critical reading is required to insure that the important
information that will lead to a logical, correct answer in a problem-solving situation be
found. Critical problem solving is required to insure that the information necessary to
solving the problem is processed accurately to insure the accuracy of an answer. In this
course being able to lay out the information in a coherent written form is essential to the
whole problem solving process. The process used in this course is summarized as the
GFESA problem solving process. GFESA breaks down into Given, Find, Equation,
Substitution, and Answer. Critical reading focuses on the Given and Find, while Critical
problem solving focuses on the Equation, substitution and answer.
Motion: The mathematical description of motion is called kinematics. In the study of
motion you will be studying speed, velocity, and acceleration. For now we will
concentrate on motion in a straight line.
Scalar and Vector Quantities: A scalar quantity is described by the magnitude (size) of
the measurement and includes units as appropriate. For example, a distance measurement
of 6 km or a speed measurement of 40 km / hr are examples of scalar quantities. On the
other hand, a vector quantity is described both by its magnitude, with appropriate units,
and by direction. For example, a displacement measurement of 2.5 km, north and a
velocity measurement of 30 km / hr, E (east) are examples of vector quantities. Note the
difference in the way scalar and vector quantities are expressed. Scalar quantities are
measurements that are most familiar to you. They involve no statement of direction. They
have only a numerical value followed by one or more units of measure. Vectors, on the
other hand, must have a statement of direction following the numerical value and its
unit(s). Statements of direction can have many forms such as forward, reverse, up, down,
a compass angle and a unit circle angle. Never ignore direction unless a question only
asks for the magnitude. In such a situation you need not report a direction in an answer.
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Vector
Addition - Graphical Method: Vectors can be combined. We usually say they
are added, though, except in the case of straight-line motion the combination of two or
more vector quantities involves more that simple arithmetic. (i.e., for motion in two
dimensions, vector quantities are combined using Trig. functions) Vector addition can be
accomplished with reasonable results using a few simple skills learned in geometry.
Equipped with a ruler and a protractor, you can achieve fairly good results when
combining (adding) vector quantities. This approach will be used in your study of
vectors, but will be replaced with mathematical solutions later on in order to save time.
Distance and Displacement: Distance (d) measurements are scalar quantities. For
example, the distance between to points found on a map are 5 km apart. Notice that no
reference is made to one or the other point as being the point from which the
measurement is made from. Also there is no apparent statement of direction in the
expression of this measurement. You could also call distance the length moved. On the
other hand, displacement (d) measurements are vector quantities. For example, the
displacement of a person after having taken a walk is 1.5 km 20o to the east of north. In
this situation you are using the person's starting point as a reference point and you are
stating the person's displacement both in terms of a distance and in terms of a direction.
Displacement measurements should always inform you of where the person is now with
respect to their starting point. In a sense distance doesn't allow for you to find them while
displacement does. Distance is normally expressed in meters (m), though kilometers are
often used too. Displacement also uses meters for the scalar portion of the measurement.
Direction is often described using angles expressed in degrees (o) on a compass or unit
circle.
Instantaneous Speed and Velocity: Speed (v) is often defined as the distance an object
travels per unit of time. Instantaneous Speed (vi or vf) is the speed an object is moving at
any given instant of time. Speed is normally expressed in meters / second (m/s). A
speedometer is frequently used to measure instantaneous speed. An example of speed is
25 km/h. Velocity (v) being a vector quantity uses a scalar measurement, which is its
magnitude, and a direction. Instantaneous Velocity (vi or vf) is the velocity an object has
at a given instant of time. It includes the measure of instantaneous velocity followed by a
statement of direction. For example 25 km/h, west is an example of an instantaneous
velocity.
Average Speed and Average Velocity: When you travel somewhere, it is highly
unlikely that you move with a constant speed or experience a constant velocity. What is
more likely to be happening is that your speed and possibly your velocity are constantly
changing. As a consequence, it is far easier to express your average speed for the entire
trip you have made than it is to keep track of all of the instantaneous velocities during the
trip. The equation for average speed is vAVE = d / t , where d is the distance traveled and t
is the time over which the distance was traveled. Average velocity is found using the
same equation vAVE = d / t, if the direction of motion is in a straight line, because only the
magnitude (size) of the velocity is changing. If the object changes direction then the
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situation
becomes more complicated and may require other equations be used. These
equations will be introduced later in the course.
Acceleration: Acceleration (a) is usually defined as the rate of change in velocity, much
as velocity is often defined as the change in position, which is another way of describing
displacement. The acceleration can be found by the following equation when travel is
along a straight line. a = (vf - vi)/ t, which tells us that acceleration is the change in
velocity divided by the time interval through which the acceleration occurs.
Calculating Final Velocity After Uniform Acceleration: After an object experiences
acceleration, it may be necessary to calculate the object's final velocity. To do this we
rearrange the equation above and end up with vf = vi + a t. What would the equation look
like if you were to solve for the initial velocity after an object experienced an
acceleration?
Calculating Displacement During Uniform Acceleration: The equation d = vAVE / t is
valid for both an object traveling with constant speed or an object experiencing uniform
(constant) acceleration. The equation can be written in another form as shown. d = vAVE /
t = (vf + vi / 2) / t
Calculating Displacement From Acceleration and Time: The equation for
displacement in this situation is d = vi t + 0.5 ( a ) t2. It allows you to calculate
displacement when all you know is the initial velocity, the rate of acceleration, and the
time interval through which the acceleration occurred.
An Equation Independent of Time: If you are looking for final velocity when you have
no information about time, you may find the following equation useful. vf2 = vi2 + 2 a d.
Calculating Acceleration From Displacement and Velocity: The above equation can
be changed around (algebraically) to solve for displacement. This form is
d = (vf2 - vi2) / 2 a.
Acceleration Due to Gravity: All objects close to the earth's surface fall downwards,
that is they accelerate downwards, at the same rate, if we do not factor in air resistance. In
vacuum chambers on earth and on the moon where there is no air resistance, because
there is no air, all objects even coins and feathers all fall at the same rate. The rate of
downwards acceleration is 9.81 m / s2 (when rounded, 9.8 m/s2) at sea level. As you
move farther from the surface this value represented by the symbol g and called the
acceleration due to gravity, decreases. You will see that this decrease in acceleration, due
to an increase in distance between the two objects, in this case the earth and the object
being accelerated downwards, is the effect distance has on the force of gravity between
the two objects. Of most significance to you in terms of problem solving is that the
equations you already learned concerning acceleration in a straight line on horizontal
surfaces also apply to falling objects. The only difference in your calculations is that you
will replace the more generic variable symbol a (acceleration in general) with the more
specific variable symbol g (the acceleration due to gravity). Always remember that g does
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NOT
represent gravity or the force of gravity. It is a symbol that represents acceleration,
acceleration that is caused by gravitational force.
Compare the following equations:
Horizontal Motion Equations
vf = vi - a t
vf2 = vi2 + 2 a d
d = vi t + 0.5 ( a ) t2
Vertical Motion Equations (falling bodies)
vf = vi - g t
vf2 = vi2 + 2 g d
d = vi t + 0.5 ( g ) t2
Vocabulary: motion diagram, operational definition, particle model, coordinate system,
origin, position vector, scalar quantity, vector quantity, displacement, time and time
interval, distance, displacement, speed, instantaneous and average velocity, instantaneous
and average acceleration, final velocity and initial velocity, uniform motion, acceleration
due to gravity, kinematics.
Skills to be learned: You should learn to solve for any of the variables concerning
straight-line motion, either horizontal or vertical, as described in the above information.
Assignments:
Textbook: Read / Study / Learn Chapter 3 Sections 3.1 and 3.2 about motion,
Section 3.3 about velocity and acceleration, and Chapter 4 references to solving
motion problems mathematically using equations as defined in table 5-2 on page
101 of the textbook.
WB Exercise(s): PS#3-1, 4-1, 4-2, 4-4 no. 1->4
Activities: TBA
Resources:
This Handout and the Overhead and Board Notes discussed in class
Textbook: Chapter 3 and Example Problems in Chapter 5 as applies to solving
motion problems using equations.
Workbook: Lessons & Problem Sets:
www.physicsphenomena.com - “Motion in a Straight Line”
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