Bài giảng vật lý đại cương chương 2

Chapter 2. Motion in 1-D
2.0. Some mathematical concepts
2.1. Position, Velocity and Speed
2.2. Instantaneous Velocity and Speed
2.3. Acceleration
2.4. One-Dimensional Motion with Constant Acceleration
2.5. Freely falling Object
2.6. Kinematic Equations Derived from Calculus


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,
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.
magnitude
• A Vector must be described with both magnitude and
direction.
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.
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 km along the beach.
(b) I walked 2 km due north along the beach.
(c) I jumped off a cliff and hit the water traveling at
25 km per hour.
(d) I jumped off a cliff and hit the water traveling
straight down at 25 km per hour.
(e) My bank account shows a negative balance of
–25 dollars.


2.1. Position, Velocity and Speed
• 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 at a
horizontal distance of 2.00
m to the left of a spot of the
sidewalk designated as the
origin.


A student is walking slowly.
Her horizontal position
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
∆x
vx ≡ vx ≡
∆t
r
r ∆x ∆x r
i
Vectorial form : v =
=
∆t ∆t
• SI Unit of Average Velocity: meter per second (m/s)

Use this
form from
now on


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 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 the figure below between
the times t1 = 1.0 s and t2 = 1.5 s.


Average Speed
Average speed is defined as:

total distance
v=
∆t


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 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?


2.2. Instantaneous Velocity and Speed
The instantaneous velocity is the derivative of the object’s position with
respect to time

r
r
∆x dx dx )
r
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
• 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?


2.3. Acceleration

Change in velocity
Average acceleration=
Elapsed time
r r
r
r v2 − v1 ∆v
a =

=
t2 − t1 ∆t

SI Unit of Average Acceleration: meter
per second squared (m/s2)


Instantaneous acceleration:

r
r
2r
d x
r dv d dx
a=
= ( )= 2
dt dt dt
dt
Acceleration is the 2nd derivative of position with respect to time


• An object is accelerated even only direction
changes (e.g. uniformly circular motion, next
chapter).
• 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?