Kinematics in One Dimension
Mechanics
Kinematics
(Chapter 2 and 3)
The movement of
an object itself
Concepts needed
to describe motion
without reference
to forces
Dynamics
(Chapter 4)
Deals with the
effect that forces
have on motion
 The
displacement of an object is a vector
that points from an object initial position
to its final position and has a magnitude
that equals the shortest
distance
between

the two positions. ∆ x
 SI Unit: meter (m)
 When motion is along a straight line,
directions can be assigned as positive or
negative
  
x  x  x
The displacement is the difference between the
final and initial position of an object. Remember
that delta means change in. The change in any
variable is always the final value minus the
initial value.
 Speed
can be described as how fast an
object is moving.
 Average speed is the distance traveled
divided by the time required to cover the
distance.
 SI unit: m/s
Distance
AverageSpeed 
ElapsedT ime
 Average
velocity is found by dividing the
displacement by the elapsed time.
 SI unit: m/s
 Average velocity is a vector that points in the
same direction as the displacement. Again, for
one dimension, + or – can be used to indicate
direction of velocity.
 

 x  x x
v

t  t
t
 The
instantaneous velocity of an object
indicates how fast the object moves and
the direction of the motion at each instant
of time.


x
v  lim
t  0 t
 What
is the difference between speed
and velocity?
 Give an example of positive velocity and
negative velocity.
 Explain how average velocity and
instantaneous velocity are different.
http://he-cda.wiley.com/WileyCDA/HigherEdMultiTitle.rdr?name=cutnell
 The
slope of a displacement vs. time
graph will tell you the velocity of the
object.
 If the slope is positive, the velocity is
positive.
 It should go without saying that if the
slope is negative, the velocity is negative
Constant
Velocity
Positive Velocity
Fast, Leftward(-)
Constant Velocity
Positive Velocity
Changing Velocity
(acceleration)
Negative (-)
Velocity
Slow to Fast
Leftward (-)
Velocity
Fast to Slow
How could we find
the instantaneous
velocity in any of
these situations?
 The
slope of a velocity vs. time graph
reveals useful information about the
acceleration of an object
 If the acceleration is zero, the slope is
zero (horizontal)
 If acceleration is positive, slope is
positive
 If the acceleration is negative, slope is
negative
http://www.physic
sclassroom.com/Cl
ass/1DKin/u1l4c.cf
m
 The
area under the curve of a velocity vs.
time graph can be used to determine the
displacement of an object.
 Acceleration
comes into discussions of
motion when the change in the velocity is
combined with the time during which the
change occurs.
 Average Acceleration  is a vector that
a
points in the same direction as 
v
 As with velocity, plus and minus signs
indicate the two possible direction for the
acceleration vector when motion is along
a straight line.
SI unit of acceleration: m/s2
Instantaneous
acceleration is a
limiting case of the
average
acceleration. When
the time interval for
measuring the
acceleration
becomes extremely
small, the average
acceleration and
the instantaneous
acceleration
become equal.
 

 v  vo v
a

t  to
t


v
a  lim
t 0 t
In most
situation we
deal with
acceleration is
constant or
“uniform.”
A drag racer crosses the finish line, and the driver
deploys a parachute and applies the brakes to slow
down. The driver begins slowing down when t = 9.0s
and the car’s velocity is +28m/s. When t = 12.0s, the
velocity has been reduced to +13m/s. What is the
average acceleration of the dragster?
 Whenever
the acceleration and velocity
vectors have opposite directions, the
object slows down and is said to be
“decelerating.”
 When the acceleration and velocity
vectors point in the same direction, the
object speeds up and has positive (+)
acceleration.
 We
have now discussed the motion of
objects along a straight line in terms of the
following quantities
•
•
•
•
Displacement
Velocity
Time
Acceleration
 The
kinematics equations use no new
concepts but relate these terms in easy to
use equations.
 These equations need only be derived one
time, then used over and over.
 Object

x0  0m
is located at the origin
when t0  0s. Then, ∆x becomes x.
 Dispense using boldface symbols and
small arrows for displacement, velocity,
and acceleration vectors.
 Continue to use + or – for direction.
 Essentially, kinematics equations
rearrange and recombine existing
equations in order to make problem
solving easier.
1. Start with
acceleration
equation and
rearrange to solve
for v.
2. Using the
average
velocity
formula and
assuming x0
and t0 to be
zero, x = vt .
v  v0
a
t
Since the
velocity
increases at a
constant rate,
the average
velocity is
midway
between v0
and v.
v  v0  at
x  vt 
1
(v0  v)t
2
x  (v0  v)t
1
2
3. Combining
equations one and two
into a single equation
by algebraically
eliminating the final
velocity (v) expression
for displacement (x) will
result.
4. Finally, by combining the
average acceleration formula
(rearranged to solve for t) and
equation number 2, an
expression of motion not
involving time can be derived.
1 2
x v 0 t  at
2
v  v  2ax
2
2
0
Draw a Picture
 Decide which direction will be positive (+) and
which will be negative (-)
 While you are reasoning the problem, be sure to
interpret the terms “decelerating” or
“deceleration” correctly
 If the motion of two objects are interrelated, data
for only two variable need to be specified for
each object.
 If motion is divided into segments, with different
acceleration, realize that final velocity for one is
initial velocity for the other.

 In
general, all bodies at the same location
above the earth fall vertically with the
same acceleration.
 When we neglect air resistance and
consider acceleration nearly constant, we
are considering “free fall” and can use
kinematics equations in problem solving
 g = 9.8m/s2
g
on the moon is approx 1/6 that of g on
Earth
 Often, the symbol “y” is used for vertical
displacement
 When an object is dropped, v0 = 0m/s
 The acceleration due to gravity is
ALWAYS a downward-pointing vector.
 Free
Fall refers to any object moving
either upward or downward under the
influence of gravity alone
 In either case, the object always
experiences the same downward
acceleration due to gravity