Chapter 2
Kinematics – Rectilinear Motion
Displacement, Velocity & Acceleration
•
Equations describing motion along straight
line
•
Objects falling freely under earth‟s gravity
•
MECHANICS
DYNAMICS
(Force & Response)
KINEMATICS
(describes motion)
STATICS
(At rest)
Concepts to study objects in motion

Position

Displacement (similar to distance)

Velocity (similar to speed)

Acceleration
What is “Position” in Physics?
Frame of reference
 A choice of coordinate axes with an
origin
 Coordinates for representing starting
and ending points of a motion
 One dimension only (this chapter)
 (example) a straight line extending to
the right and left of the point at the
center (origin) and called the x
direction
Displacement

Defined as the change in position
 x  x  x
f
i


f - final position; i - initial
Units -meters (SI)
CAN BE POSITIVE OR
NEGATIVE ALONG A
GIVEN DIRECTION !!!


From A to B
 xi = 30 m
 xf = 52 m
 x = 22 m
 The displacement is
positive, indicating
the motion was in the
positive x direction
From C to F
 xi = 38 m
 xf = -53 m
 x = -91 m
 The displacement is
negative, indicating
the motion was in the
negative x direction
Some examples of displacement
Sign of ∆x
Initial x
Final x
∆x
30
52
22
+
38
-53
-91
-
-50
-30
20
+
42
31
-11
-
Displacement is a “vector”


(Need both magnitude (size) and
direction to completely describe)
For 1 D motion just + or – sign
(Scalar quantities have just magnitude)
IMPORTANT NOTE
Motion along one dimension may be either
along x or y or z axis direction and can be
used to represent horizontal side to side
(motion along x axis) or front and back
(motion along z axis) or vertical up and down
motion (motion along y axis)
Displacement not always ≠ distance

Throw a ball straight up and then catch it
at the same point you released it
Distance = 2 times height reached by ball

The displacement is zero!!!!!!
x x
i
f
so
x
is zero
Two dimensions Example
East
Walk 3 blocks east and then 4 blocks north
 Total distance = ?
 Displacement = ?
Speed

The average speed of an object is defined
as the total distance traveled divided by
the total time elapsed
total distance
Average speed 
total time
d
v 
t



SI units – meters/second m/s
Speed - scalar? Vector?
Speed always positive?
You drove to Trenton campus 8 miles.
You drove for 17 minutes and stopped
for 3 mins. at a gas station on your
way. What was your :
Displacement?
Distance covered?
Average speed?



+8 miles relative to origin at WWC
8 miles
0.4 miles/min or 10.7 m/s
Think!

What is the displacement for the round
trip if you returned back to college?
Velocity
Average velocity =
vaverage
displacement
time elapsed
x xf  xi


t
tf  ti
SI units – m/s (other units feet/s, cm/s)
Velocity can be positive or negative
Velocity is a vector! For 1D, just +ve or
negative
Direction of velocity same as displacement
You drove to Trenton campus 8 miles.
You drove for 17 minutes and stopped
for 3 mins. at a gas station on your way.
What was your average velocity?
Average velocity =+ 10.7 m/s
What is the average velocity for the
round trip?
Speed vs. Velocity

average velocity = average speed?
Note: In text, bars over physical
quantities represent average values.
e. g. average velocity, average speed,
average acceleration
Quick Quiz 2.1
Total distance =
Displacement =
Average velocity in x direction =
Average Speed =
200 yards, 0, 0, 200÷25 = 8 yards/sec
Chapter 2 Lecture 2
understand changing velocity & acceleration;
motion with constant acceleration
special case of constant acceleration: free fall
Understanding Velocity & Acceleration
from Graphs




Position(x) vs. time (t) & velocity(v) vs.
time (t) graph
Object moving with a constant velocity
(uniform motion) – what is the shape of x –t
graph?
Average velocity = slope of line joining
initial & final positions
Slope of a line is rise divided by run
change in vertical axis
slope 
change in horizontal axis
comparing constant & changing velocity
equal displacements in
equal intervals of time;
average velocity is
constant all the time;
uniform motion
average velocity (slope of
curve) changes from time
to time; change in velocity
means acceleration!
Instantaneous velocity


For non uniform velocity, use instantaneous
velocity (called simply v) instead of vavg
The limit of the average velocity as the time
interval becomes infinitesimally short, or as
the time interval approaches zero
v 

lim
t  0
x
t
instantaneous velocity indicates what is
happening at every point of time when
velocity keeps changing
Instantaneous Velocity on a Graph

The slope of the line tangent to the
position-vs.-time graph is defined to be
the instantaneous velocity at that time

The instantaneous speed is defined as the
magnitude of the instantaneous velocity
Acceleration
change in velocity
Acceleration 
time elapsed
v vf  vi
a

t
tf  ti



This is average acceleration
Units are m/s² (SI), cm/s² (cgs),
and ft/s² (US Customary)
Vector? Yes, in 1D + or -
What are vi and vf ?
Instantaneous values of velocities (initial and
final)
Class Example 1
A car takes 2 seconds to accelerate from an
initial velocity of +10 m/s to a final velocity
+20 m/s. What is the average acceleration?
ā =+5 m/s2
acceleration is positive and in the same
direction as initial velocity
Class Example 2
A plane travelling at +30 m/s on the
runway stops in 5 seconds. What is the
acceleration?
ā = -6 m/s2
(a is negative and it has a sign
opposite to that of initial velocity)
TRUE OR FALSE
“So, to slow down an object you always
need to have negative acceleration. To
speed up an object, you always need to
have positive acceleration.”
FALSE, FALSE, FALSE, FALSE, FALSE, FALSE
Example: Motion Diagram of A Car
Please insert active
figure 2.12
Average Acceleration


When the sign of the initial velocity and the
acceleration are the same (either positive or
negative), then the speed is increasing
When the sign of the initial velocity and the
acceleration are in the opposite directions,
the speed is decreasing
Negative Acceleration


A negative acceleration does not
necessarily mean the object is
slowing down
If the acceleration and velocity are
both negative, the object is
speeding up
Finding Average Acceleration From v-t graph
Relationship Between Acceleration and
Velocity


Uniform velocity (shown by red arrows
maintaining the same size)
Acceleration equals zero
Relationship Between
Velocity and Acceleration




Velocity and acceleration are in the same
direction
Acceleration is uniform (blue arrows maintain
the same length)
Velocity is increasing (red arrows are getting
longer)
Positive velocity and positive acceleration
Relationship Between
Velocity and Acceleration




Acceleration and velocity are in opposite
directions
Acceleration is uniform (blue arrows maintain
the same length)
Velocity is decreasing (red arrows are getting
shorter)
Velocity is positive and acceleration is
negative
Quick Quiz 2.2
True or False?
a)
b)
c)
a.
b.
c.
A car must always have an acceleration in the
same direction as velocity
It is possible for a slowing car to have +ve
acceleration
An object with constant nonzero acceleration can
stop but then can not remain at rest
Answers:
Think about class example 2
Think about slowing car with initial velocity of -10
m/s and final velocity -5 m/s
Think about +ve velocity, -ve acceleration of chalk
thrown vertically upward
1D motion with const. acceleration
Equations with constant acceleration, a
v  vo  at
1
x  vt  vo  v  t
2
1
x  vot  at 2
2
v 2  vo2  2ax
Objects thrown up or down on earth move with
a special constant acceleration called “g” the
acceleration due to gravity = 9.8 m/s2
Notes on the equations
 If initial t=0 & final t=t, then ∆t = t
x  v av erage


 vo  vf
t
 2

t

Gives displacement as a function of
velocity and time
Use when you don‟t know and
aren‟t asked for the acceleration
Notes on the equations
v  vo  at


Shows velocity as a function of
acceleration and time
Use when you don‟t know and
aren‟t asked to find the
displacement
Notes on the equations
1 2
x  v o t  at
2


Gives displacement as a function
of time, velocity and acceleration
Use when you don‟t know and
aren‟t asked to find the final
velocity
Notes on the equations
2
2
o
v  v  2ax


Gives velocity as a function of
acceleration and displacement
Use when you don‟t know and
aren‟t asked for the time
Textbook Example 2.4, page 37, Daytona 500
Starts from rest: at t=0, v is zero (v0 = 0)
Constant acceleration: uniform a = 5m/s2
Velocity of car after a distance of 100 ft =?
Use equation:
v  v  2ax
2
2
0
For distance convert feet to m! (1 m = 3.281 feet)
Time elapsed=?
Use equation:
v  v 0  at
Textbook example 2.4 Ans: 17.5 m/s & 3.50 s
End of Chapter MC1:
(!!! sig fig & units!!!)
Choose upward vertical direction as +ve y axis
(coming up: free fall, use „y‟ axis)
Initial velocity +15
Final velocity -8
(Assume a = -9.8 ….coming up)
Find t
Use
v
 at
v
0
2.35 s
Problem 9
Instantaneous velocities of tennis player at
a) 0.50 s
b) 2.0 s
c) 3.0 s
d) 4.5 s
Just find slope of graph around each t value
4, -4, 0, 2 with sig fig & units
Galileo Galilei



1564 - 1642
Galileo formulated
the laws that govern
the motion of objects
in free fall
Also looked at:




Inclined planes
Relative motion
Thermometers
Pendulum
Free Fall

All objects moving under the influence
of gravity only are said to be in free fall



Free fall does not depend on the object‟s
original motion
All objects falling near the earth‟s
surface fall with a constant acceleration
The acceleration is called the
acceleration due to gravity, and
indicated by g
Acceleration due to Gravity


Symbolized by g
g = 9.80 m/s² pointing towards the earth


g is always directed downward



g 10 m/s2
Toward the center of the earth
Ignoring air resistance and assuming g
doesn‟t vary with altitude over short vertical
distances, free fall is constantly accelerated
motion
Who is in free fall?
Space shuttle!!!
Free Fall – an object
dropped



Initial velocity is
zero
Let up be positive
Use the kinematic
equations
Tip: Generally use y
instead of x since
vertical

Acceleration is g
= -9.80 m/s2
vo= 0
a=g
Free Fall – an object
thrown downward


a = g = -9.80
m/s2
Initial velocity  0

With upward
being positive,
initial velocity will
be negative
Free Fall -- object thrown
upward



Initial velocity is
upward, so positive
The instantaneous
velocity at the
maximum height is
zero
a = g = -9.80 m/s2
everywhere in the
motion
v=0
Thrown upward, cont.

The motion may be symmetrical



Then tup = tdown
Then v = -vo
The motion may not be
symmetrical

Break the motion into various parts

Generally up and down
Problem 45: thrown upward with speed 25
This is free fall motion with uniform a=g=9.8
Upward direction is +ve y axis & use g=-9.8
a) Maximum height reached
At this value of ∆x, final v = 0; initial v =
25
2
2
Use
  2ay
v v
b) Use
0
v  v 0  at
31.9m, 2.55s, same t, same initial v & opposite
Class Example
Obama throws a rock down with speed of 12 m/s
from the top of a tower. The rock hits the ground
after 2.0s. What is the height of the tower?
Choose down as positive & choose ∆y for displacement
Given initial velocity (v0=12)
Given time (t = 2)
Given free fall (moving under constant downward
acceleration (a=g=9.8)
Asked to find ∆y - the distance traveled by rock from
top to bottom (height of tower)
Use
1
2
y  v 0  t   a  t
2
44 m (rounded)

In free fall motion of chalk, there is
constant acceleration called „g‟ of
magnitude 9.8 meter per second
squared throughout its trip from the
time it leaves my hands until it hits
the ground
Non-symmetrical
Free Fall


Need to divide the
motion into
segments
Possibilities include


Upward and
downward portions
The symmetrical
portion back to the
release point and then
the non-symmetrical
portion