Chapter 2 Motion Along a Straight Line
2-0. Mathematical Concept
2.1. What is Physics?
2.2. Motion
2.3. Position and Displacement
2.4. Average Velocity and Average Speed
2.5. Instantaneous Velocity and Speed
2.6. Acceleration
2.7. Constant Acceleration: A Special Case
2.8. Another Look at Constant Acceleration
2.9. Free-Fall Acceleration
2.10. Graphical Integration in Motion Analysis
Trigonometry
Example 1 Using Trigonometric Functions
On a sunny day, a tall building casts a shadow that is
67.2 m long. The angle between the sun’s rays and the
ground is =50.0°, as Figure 1.6 shows. Determine the
height of the building.
Trigonometric Functions
PYTHAGOREAN THEOREM
h  h0  ha
2
2
2
h0
sin  
h
h0
  sin ( )
h
ha
cos  
h
ha
  cos ( )
h
h0
tan  
ha
h0
  tan ( )
ha
1
1
1
Example 1 Using Trigonometric Functions
On a sunny day, a tall
building casts a shadow
that is 67.2 m long. The
angle between the sun’s
rays and the ground is
=50.0°, as Figure 1.6
shows. Determine the
height of the building.
What is the location of downtown Wilmington?
N̂
Market St. is 6°
north of east
Ê
Ŵ
To specify a location downtown,
it’s more convenient to use the
Market St./ 3rd St. coordinate system
than the East/North coordinate
system
Ŝ
Defining a Coordinate System
One-dimensional coordinate system consists of:
• a point of reference known as the origin (or zero point),
• a line that passes through the chosen origin called a
coordinate axis, one direction along the coordinate axis,
chosen as positive and the other direction as negative,
and the units we use to measure a quantity
Scalars and Vectors
• A scalar quantity is one that can be described with a
single number (including any units) giving its magnitude.
• A Vector must be described with both magnitude and
direction.
A vector can be represented by an
arrow:
•The length of the arrow represents
the magnitude (always positive) of
the vector.
•The direction of the arrow represents
the direction of the vector.
A component of a vector along an axis
(one-dimension)
A UNIT VECTOR FOR
A COORDINATE AXIS
is a dimensionless
vector that points in the
direction along a
coordinate axis that is
chosen to be positive.
A one-dimensional vector can be constructed by:
•Multiply the unit vector by the magnitude of the vector
•Multiply a sign: a positive sign if the vector points to the same
direction of the unit vector; a negative sign if the vector points to
the opposite direction of the unit vector.
A component of a vector along an axis=sign × magnitude
Difference between vectors and scalars
• The fundamental distinction between
scalars and vectors is the characteristic of
direction. Vectors have it, and scalars do
not.
• Negative value of a scalar means how
much it below zero; negative component
of a vector means the direction of the
vector points to a negative direction.
Check Your Understanding 1
Which of the following statements, if any,
involves a vector?
(a) I walked 2 miles along the beach.
(b) I walked 2 miles due north along the beach.
(c) I jumped off a cliff and hit the water traveling at
17 miles per hour.
(d) I jumped off a cliff and hit the water traveling
straight down at 17 miles per hour.
(e) My bank account shows a negative balance of
–25 dollars.
Motion
• The world, and
everything in it, moves.
• Kinematics: describes
motion.
• Dynamics: deals with
the causes of motion.
One-dimensional position vector
• The magnitude of the position vector is a scalar that
denotes the distance between the object and the origin.
• The direction of the position vector is positive when the
object is located to the positive side of axis from the origin
and negative when the object is located to the negative
side of axis from the origin.
Displacement
• DISPLACEMENT is defined as the change of an object's
position that occurs during a period of time.
• The displacement is a vector that points from an object’s
initial position to its final position and has a magnitude
that equals the shortest distance between the two
positions.
• SI Unit of Displacement: meter (m)
Example 2: Determine the displacement in the following cases:
(a) A particle moves along a line from
to
(b) A particle moves from
to
(c) A particle starts at 5 m, moves to 2 m, and then returns to 5 m
EXAMPLE 3: Displacements
Three pairs of initial and final positions along
an x axis represent the location of objects
at two successive times: (pair 1) –3 m, +5
m; (pair 2) –3 m, –7 m; (pair 3) 7 m, –3 m.
• (a) Which pairs give a negative
displacement?
• (b) Calculate the value of the displacement
in each case using vector notation.
Velocity and Speed
A student standing still with
the back of her belt at a
horizontal distance of 2.00
m to the left of a spot of the
sidewalk designated as the
origin.
A student starting to walk
slowly. The horizontal
position of the back of her
belt starts at a horizontal
distance of 2.47 m to the
left of a spot designated as
the origin. She is speeding
up for a few seconds and
then slowing down.
Average Velocity
Displacement
Average velocity=
Elapsed time
x2  x1
x x
v 

i 
i
t t
t2  t1
•
x2 and x1 are components of the position vectors at the
final and initial times, and angle brackets denotes the
average of a quantity.
• SI Unit of Average Velocity: meter per second (m/s)
Example 4 The World’s Fastest Jet-Engine Car
Figure (a) shows that the car
first travels from left to right
and covers a distance of 1609
m (1 mile) in a time of 4.740 s.
Figure (b) shows that in the
reverse direction, the car
covers the same distance in
4.695 s. From these data,
determine the average
velocity for each run.
• Example 5: find the average velocity for
the student motion represented by the
graph shown in Fig. 2-9 between the
times t1 = 1.0 s and t2 = 1.5 s.
Average Speed
Average speed is defined as:
Check Your Understanding
A straight track is 1600 m in length. A
runner begins at the starting line, runs due
east for the full length of the track, turns
around, and runs halfway back. The time
for this run is five minutes. What is the
runner’s average velocity, and what is his
average speed?
EXAMPLE 6
You drive a beat-up pickup truck along a straight road for
8.4 km at 70 km/h, at which point the truck runs out of
gasoline and stops. Over the next 30 min, you walk
another 2.0 km farther along the road to a gasoline
station.
• (a) What is your overall displacement from the
beginning of your drive to your arrival at the station?
• (b) What is the time interval from the beginning of your
drive to your arrival at the station? What is your average
velocity from the beginning of your drive to your arrival
at the station? Find it both numerically and graphically.
Suppose that to pump the gasoline, pay for it, and walk
back to the truck takes you another 45 min. What is your
average speed from the beginning of your drive to your
return to the truck with the gasoline?
Instantaneous Velocity and Speed
x dx dx
v  lim

 i
dt dt
t 0 t
• The instantaneous velocity of an object can be obtained
by taking the slope of a graph of the position component
vs. time at the point associated with that moment in time
• The instantaneous velocity can be obtained by taking a
derivative with respect to time of the object's position.
• Instantaneous speed, which is typically called simply
speed, is just the magnitude of the instantaneous
velocity vector,
Example 7
The following equations give the position component,
x(t), along the x axis of a particle's motion in four
situations (in each equation, x is in meters, t is in
seconds, and t > 0): (1) x = (3 m/s)t – (2 m);
(2) x = (–4 m/s2)t2 – (2 m); (3) x = (–4 m/s2)t2;
(4) x = –2 m.
• (a) In which situations is the velocity of the particle
constant?
• (b) In which is the vector pointing in the negative x
direction?
How to Describe Change of Velocity ?
Definition of Acceleration
Change in velocity
Average acceleration=
Elapsed time
v2  v1 v
a 

t2  t1 t
SI Unit of Average Acceleration: meter
per second squared (m/s2)
Instantaneous acceleration:
2
dv d dx
d x
a
 ( ) 2
dt dt dt
dt
• An object is accelerated even if all that changes
is only the direction of its velocity and not its
speed.
• It is important to realize that speeding up is not
always associated with an acceleration that is
positive. Likewise, slowing down is not always
associated with an acceleration that is negative.
The relative directions of an object's velocity and
acceleration determine whether the object will
speed up or slow down.
EXERCISE
A cat moves along an x axis. What is the sign of
its acceleration if it is moving
(a) in the positive direction with increasing speed,
(b) in the positive direction with decreasing speed,
(c) in the negative direction with increasing speed,
and
(d) in the negative direction with decreasing
speed?
EXAMPLE 7: Position and Motion
A particle's position on the x axis of Fig. 2-1
is given by
with x in meters and t in seconds.
• (a) Find the particle's velocity function and
acceleration function .
• (b) Is there ever a time when vx  0 ?
• (c) Describe the particle's motion for t  0
Constant Acceleration: A Special Case
Free-Fall Acceleration
Equations of Motion with Constant Acceleration
v2 x  v1x  ax t
1
x  (v1x  v2 x )t
2
1
2
x  v1x t  ax t
2
2
2
v2 x  v1x  2ax x
Example 8 A Falling Stone
A stone is dropped from
rest from the top of a
tall building, as Figure
2.17 indicates. After
3.00 s of free-fall,
(a) what is the velocity of
the stone?
(b) what is the
displacement y of the
stone?
Example 9 An Accelerating Spacecraft
The spacecraft shown in
Figure 2.14a is traveling with
a velocity of +3250 m/s.
Suddenly the retrorockets
are fired, and the spacecraft
begins to slow down with an
acceleration whose
magnitude is 10.0 m/s2.
What is the velocity of the
spacecraft when the
displacement of the craft is
+215 km, relative to the point
where the retrorockets began
firing
Example 10
Spotting a police car, you brake your Porsche
from a speed of 100 km/h to a speed of 80.0
km/h during a displacement of 88.0 m, at a
constant acceleration.
• What is that acceleration?
• (b) How much time is required for the given
decrease in speed?
Graphical
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Conceptual Question
1. A honeybee leaves the hive and travels 2 km before
returning. Is the displacement for the trip the same as
the distance traveled? If not, why not?
2. Two buses depart from Chicago, one going to New York
and one to San Francisco. Each bus travels at a speed
of 30 m/s. Do they have equal velocities? Explain.
3. One of the following statements is incorrect. (a) The car
traveled around the track at a constant velocity. (b) The
car traveled around the track at a constant speed.
Which statement is incorrect and why?
4. At a given instant of time, a car and a truck are
traveling side by side in adjacent lanes of a highway.
The car has a greater velocity than the truck. Does
the car necessarily have a greater acceleration?
Explain.
5. The average velocity for a trip has a positive value. Is
it possible for the instantaneous velocity at any point
during the trip to have a negative value? Justify your
answer.
6. An object moving with a constant acceleration can
certainly slow down. But can an object ever come to a
permanent halt if its acceleration truly remains
constant? Explain.