MOTION IN ONE

DIMENSION
Displacement,
Velocity, and
Acceleration
 Definition of Terms
▪ Motion is the activity or process of continually
changing position or moving from one place to
another
▪ Dynamics is the study of motion and of physical
concepts such as force and mass
▪ Kinematics is a part of dynamics that describes
motion without regard to its causes
 Definition of Terms
▪ Displacement is the change in position (vector
quantity)
▪ Distance is the magnitude of the displacement
(scalar quantity)
▪ A frame of reference is a choice of coordinate
axes that defines the starting point for measuring
any quantity.
 Displacement
The displacement ∆𝑥 of an object is defined as its
change in position and is given by

∆𝒙 = 𝒙𝒇 − 𝒙𝒊
𝑥𝑖 - initial position
𝑥𝑓 - final position
 Average Speed
The average speed of an object over a given time
interval is the length of the path it travels divided by
the total elapsed time:

𝒑𝒂𝒕𝒉 𝒍𝒆𝒏𝒈𝒕𝒉, (𝒎)

𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒔𝒑𝒆𝒆𝒅, (𝒗) =
𝒆𝒍𝒂𝒑𝒔𝒆𝒅 𝒕𝒊𝒎𝒆, (𝒔)
 Average Velocity
The average velocity during a time interval ∆𝑡 is the
displacement ∆𝑥 divided by ∆𝑡:

∆𝒙 𝒙𝒇− 𝒙𝒊
ഥ=
𝒗 =
∆𝒕 𝒕𝒇− 𝒕𝒊
 Average Acceleration
The average acceleration during the time interval
∆𝑡 is the change in velocity ∆𝑣 divided by ∆𝑡:

∆𝒗 𝒗𝒇− 𝒗𝒊
ഥ=
𝒂 =
∆𝒕 𝒕𝒇− 𝒕𝒊
 Average Acceleration
▪ When the object’s velocity and acceleration are in
the same direction, the speed of the object
increases with time.

▪ When the object’s velocity and acceleration are in

the opposite direction, the speed of the object
decreases with time.
 Average Acceleration
▪ The minus sign indicates that the acceleration
vector is also in the negative x-direction. Positive
and negative accelerations specify directions
relative to chosen axes. The terms speeding up
or slowing down refer to an increase and a
decrease in speed, respectively.
 Average Acceleration
▪ Negative Acceleration doesn’t necessarily mean
an object is slowing down. If the acceleration is
negative and the velocity is negative, the object is
speeding up.

▪ Deceleration means a reduction in speed, a

slowing down.
One-Dimensional
Motion Constant
Acceleration
 Constant Acceleration
When an object moves with constant acceleration,
the instantaneous acceleration at any point in a
time interval is equal to the value of the average
acceleration over the entire time interval.
Consequently, the velocity increases or decreases
at the same rate throughout the motion.
Constant Acceleration

𝒗 = 𝒗𝟎 + 𝒂𝒕
𝟏 𝟐
∆𝒙 = 𝒗𝟎 𝒕 + 𝒂𝒕
𝟐
𝟐 𝟐
𝒗 = 𝒗𝟎 + 𝟐𝐚∆𝒙
 Sample Problem 1
A race car starting from rest
accelerates at a constant rate of
5.00 m/s2. (a) What is the velocity
of the car after it has traveled 1.00
x 102 ft? (b) How much time has
elapsed?
 Sample Problem 2
A Cessna aircraft has a liftoff speed of
120 km/h. (a) What minimum constant
acceleration does the aircraft require if
it is to be airborne after a takeoff run of
240 m? (b) How long does it take the
aircraft to become airborne?
 Sample Problem 3
A truck covers 40.0 m in 8.50 s
while uniformly slowing down to a
final velocity of 2.80 m/s. (a) Find
the truck’s original speed. (b) Find
its acceleration.
Freely Falling
Objects
 Freely Falling Objects
When air resistance is negligible, all objects dropped
under the influence of gravity near Earth’s surface
fall toward Earth with the same acceleration.

A freely falling object is any object moving freely

under the influence of gravity alone, regardless of
its initial motion.
Symmetries in Free Fall
Time Symmetry Speed Symmetry
acceleration due to gravity
𝟐
𝒈 = −𝟗. 𝟖 𝒎/𝒔
 Sign Conventions
Height
▪ 𝑦 is positive (+) if the object is above the origin
▪ 𝑦 is negative (–) if the object is below the origin
Origin is the point of release

▪ Velocity
▪ 𝑣 is positive (+) if the object is moving upward
▪ 𝑣 is negative (–) if the object is moving downward
 Maximum Height
▪ At maximum height, the final velocity of an object
moving upward is equal to zero.

▪ The time it takes for an object to reach its

maximum height is expressed as:
𝒗𝟎
𝒕𝒎𝒂𝒙 = −
𝒈
Freely-Falling Bodies

𝒗 = 𝒗𝟎 + 𝒈𝒕
𝟏 𝟐
∆𝒚 = 𝒗𝟎 𝒕 + 𝒈𝒕
𝟐
𝟐 𝟐
𝒗 = 𝒗𝟎 + 𝟐𝒈∆𝒚
 Sample Problem 4
A ball is thrown vertically upward with a
speed of 25.0 m/s. (a) How high does it rise?
(b) How long does it take to reach its highest
point? (c) How long does the ball take to hit
the ground after it reaches its highest point?
(d) What is its velocity when it returns to the
level from which it started?
 Sample Problem 5
A ball is thrown directly downward
with an initial speed of 8.00 m/s,
from a height of 30.0 m. After what
time interval does it strike the
ground?
 Sample Problem 6
A construction worker accidentally
drops a brick from a high scaffold.
a) What is the velocity of the brick
after 4.5 s? b) How far does the
brick fall during this time?
 Activity 4
[p56/#27] An object moving with
uniform acceleration has a velocity of
12.0 cm/s in the positive x-direction
when its x-coordinate is 3.00 cm. If its
x-coordinate 2.00 s later is –5.00 cm,
what is its acceleration
 Activity 4
[p56/#33] In a test run, a certain car
accelerates uniformly from zero to 24.0 m/s
in 2.95 s. (a) What is the magnitude of the
car’s acceleration? (b) How long does it take
the car to change its speed from 10.0 m/s to
20.0 m/s? (c) Will doubling the time always
double the change in speed? Why?
 Activity 4
[p57/#47] A certain freely falling object,
released from rest, requires 1.50 s to
travel the last 30.0 m before it hits the
ground. (a) Find the velocity of the object
when it is 30.0 m above the ground. (b)
Find the total distance the object travels
during the fall.
 Activity 4
[p57/#50] A small mailbag is released
from a helicopter that is descending
steadily at 1.50 m/s. After 2.00 s, (a) what
is the speed of the mailbag, and (b) how
far is it below the helicopter? (c) What are
your answers to parts (a) and (b) if the
helicopter is rising steadily at 1.50 m/s?
 End of
Lesson 2
Do you have any questions?
CREDITS: This presentation template was
created by Slidesgo, including icons by
Flaticon, infographics & images by Freepik