Kinematics
Kinematics
 study of motion disregarding the forcesinvolved.
2 Kinds of Motion
1. Uniform Motion
– motion along a straight path at constant speedor velocity.
vt
s
=
2. Uniformly Accelerated Motion
– motion along a straight path at constant acceleration with variable velocities.
at
v
v
+
=
0
(
29
0
2
0
2
2
x
x
a
v
v

+
=
2
0
0
2
1
at
t
v
x
x
+
=

Note:
a positive sign is affixed on the value of a if the velocity is increasing by a uniform rate and a negative sign is affixed
on the value of a if the velocity decreases
by a uniform rate.
Displacement
 a vector quantity that represents the straightline distance between the starting and end points.
Average velocity
– the total displacement of an object divided by the elapsedtime.
)
(
1
2
1
2
motion
line
straight
t
x
t
t
x
x
V
av
∆
∆
=


=
Graph of an object’s position as a function of time, xt graph
The path in the figure does not represent the objects path in space, rather it is straight line.
It is a pictorial way to represent
how the object’s position changes
with time
Instantaneous velocity
∆t
∆x
Slope = average velocity
t
x
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The averagevelocity of a particle during a time interval can not tell us how fast or in what direction the particle was moving
at any given time during the interval.
To describe the motion in greater detail, we need to define the velocity at any specific
instant of time or specific point along the path.
Such velocity is called instantaneous velocity.
)
(
lim
0
motion
line
straight
dt
dx
t
x
V
t
=
∆
∆
=
→
∆
The limit of
t
x
∆
∆
as
t
∆
approaches zero is called the derivative of x with respect to t is equal to
dt
dx
.
The instantaneous
velocity is the limit of the averagevelocity as the time interval approaches
zero.
(a)
(b)
(c)
(a & b) as the average velocity is calculated over shorter
time intervals, its value approaches
the instantaneous velocity.
(c) in the limit
0
→
∆
t
the slope of the line
2
1
P
P
approaches
the slope of the tangent to the
t
x

curve at point
1
P
Average and Instantaneous Acceleration
Average acceleration – a vector quantity whose xcomponent is
v
∆
, the change in the xcomponent of velocity, divided by
the time interval
t
∆
)
(
1
2
1
2
motion
line
straight
t
v
t
t
v
v
a
av
∆
∆
=


=
Instantaneous Acceleration – it is the limit of the average acceleration as the time interval approaches zero.
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 Spring '11
 darp
 Calculus, Derivative, Acceleration, Force, Velocity, Aircraft carrier, XO

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