Kinematics-1
 The
study of HOW objects move:
Graphs
Equations
Motion maps
Verbal descriptions
Overview
Kinematics:
Motion
Description of
Position and displacement
velocity
»average
»instantaneous
acceleration
»average
»instantaneous
10
Position vs Time Plots
 Gives
location at any time.
 Displacement is change in position.
 Slope gives velocity.
x (m)
Write a description of this motion.
The object accelerates for the first 1
second, moving 3 m in that time. Then
it slows down until its velocity is 0 at
the 4 m mark. At the 4 m mark, it
begins to speed up in the negative
direction until it is at the 2 m mark at
2 s.
3
4
t
-3
12
Position vs Time Plots
 Gives
location at any time.
 Displacement is change in position.
 Slope gives velocity.
x (m)
Position at t=3, x(3) = 1
3
Displacement between t=1 and t=3.
4
Dx = 1.0 m - 2.0 m = -1.0 m
Average velocity between t=1 and t=3
v=
t
-3
-1 m / 2 s = -0.5 m/s
12
Instantaneous and Average
 Instantaneous
Speed - speed at any given
instant in time; speedometer reading, for
example
 Average Speed – total distance divided
by total time of travel; rate you would
have to travel constantly to cover the
same distance in the same time
Velocity vs Time Plots
 Gives
velocity at any time.
 Area gives displacement
 Slope gives acceleration.
v (m/s)
3
1.5
Velocity at t=2, v(2) = 3 m/s
Displacement between t=0 and t=3:
t=0 to t=1: ½ (3m/s) (1 s) = 1.5 m
t=1 to t=3: (3m/s) (2 s) = 6 m
7.5 m
6
4
t
-3
Average velocity between t=0 and t=3? 7.5 m / 3s = 2.5 m/s
Change in v between t=3 and t=5. Dv = -2 m/s – 3 m/s = -5 m/s
Average acceleration between t=5 and t=3:
a = -5 m/s / (2 s) = -2.5 m/s2
18
Velocity vs Time Plots
 Gives
velocity at any time.
 Area gives displacement
 Slope gives acceleration.
v (m/s)
3
Write a description of this motion.
The object accelerates from 0 m/s to
3 m/s during the first second. Then
for 2 seconds, it travels at a
constant velocity of 3 m/s. From 3
seconds to 5 seconds, it accelerates
in the negative direction from 3 m/s
to –2 m/s.
1.5
6
4
t
-3
18
Acceleration vs Time Plots
 Gives
acceleration at any time.
 Area gives change in velocity
a (m/s2)
Acceleration at t=4, a(4) =
3
-2 m/s2
Change in v between t=4 and t=1. Dv =
6
+4 m/s
24
t=1-3: Dv = (3m/s2)(2s) = 6 m/s
t=3-4: Dv = (-2m/s2)(1s) = -2 m/s
t
-3
21
Example
x (meters)
100
0
-100
position vs. time
-200
-300
0
v (m/s)
5
10
t (seconds)
15
20
•
•
•
•
Where is velocity zero?
Where is velocity positive?
Where is velocity negative?
Where is speed largest?
• Where is acceleration zero?
• Where is acceleration positive?
20
0
-20
-40
velocity vs. time
-60
-80
-100
0
5
10
t (seconds)
15
20
48
Summary of Concepts
 kinematics:
A description of motion
 position: your coordinates
 displacement: Dx = change of position
 velocity: rate of change of position
average : Dx/Dt
instantaneous: slope of x vs. t
 acceleration:
rate of change of velocity
average: Dv/Dt
instantaneous: slope of v vs. t
50
Velocity
 Slope
of x vs t gives v vs t x(t)
4s
 Area under v vs t gives Dx!
 Which
plot best represents v(t)
v(t)
v(t)
t
Dx
Dt
5s
t
v(t)
t
t
Acceleration (m/s2)

The average acceleration is the change in
velocity divided by the change in time.
Dv
a
Dt
v(t)
Dv
Dt

Instantaneous acceleration is limit of
average velocity as Dt gets small. It is the
slope of the v(t) plot
t
v(t)
t
31
Speeding Up & Slowing Down
•Negative acceleration can mean speeding
up or slowing down. The same is true with
positive acceleration.
Instantaneous Velocity

Instantaneous velocity is the limit of the average velocity
as the time interval approaches 0.
Dx dx

v  lim

Dt 0 Dt
dt


It is the instantaneous rate of change of position with
time.
Instantaneous rate of change is the derivative – which
means slope of a graph at a point
Acceleration

Acceleration is a vector quantity
defined as the rate at which an object
changes its velocity. An object is
accelerating if it is changing its
velocity.
Acceleration
Slope of V vs T – 1
Time (s) Velocity(m/s)
0
10
1
10
2
10
3
10
4
10
5
10
Acceleration
Slope of V vs T -- 2
Time (s) Velocity(m/s)
0
0
1
10
2
20
3
30
4
40
5
50
Constant Positive Velocity
Constant Negative Velocity
Position – Time
Constant Acceleration
•
Positive Velocity
Positive Acceleration
Positive Velocity
Negative Acceleration
Negative Velocity
Negative Acceleration
Negative Velocity
Positive Acceleration
Passing Lane
 When
does the red car pass the blue car?
 Where
 How
does the red car pass the blue car?
fast is each traveling at that time?
Passing Lane Velocity-Time
Passing Lane – Position-Time