chapter 2 kinematics in one dimension
DESCRIPTION
Chapter 2 Kinematics in one Dimension. September 17, 2014. Going around in circles …. . s=pathlength. A. x. A. . B. x=displacement. B. t=time interval (A,B) = 10 seconds. Seconds. . s. A. x. . B. DEFINITIONS. s=path length or total distance traveled. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 2Kinematics in one Dimension
April 24, 2023
Going around in circles …
A
B
s=pathlength
Seconds
x=displacement
xA
B
t=time interval (A,B) = 10 seconds
ABs
x
DEFINITIONS
s=path length or total distance traveled.
x=displacement = NET distance traveled. Difference between final and initial position independent of the path.
t=time of the trip. Usually measured in seconds.
More Definitions
average speed = distance traveled (s) / time taken [meters/sec.]
average velocity= displacement/time [m/s]
Note: the actual SYMBOL used for distance and displacement will usually change with the context of the problem.
smsm
txvelocity /5.5
1055~
10s
-40m
m/s 41040
sm
txvelocity
In the limit -
x
t txvavg
dtdx
txLimv
In the limit of t0, velocity is the slope of the tangent to the x-t curve!
vtxx
vtxx
vdtdx
vdtdxdtdxv
if
if
t
t
x
x i
f
i
0
For v=constant
Constant Velocity
v
xi
xf
x=0
xf – xi= distance traveled = v t
MOTION DOESN’T ALWAYS START AT THE ORIGIN
Question
253 2 ttx
The position of a particle as a function of time is given by theequation below.
(a) What is the position of the particle at t=2 seconds?(b) What is the velocity of the particle at t= 2 seconds?(c) At what time is the particle’s velocity = 0?(d) Is the particle ever at the origin? When?
Conclusion
Velocity is not the same at all times. It is changing A changing velocity is called an acceleration.
dtdva
tv
tvv
a initialfinalaverage
Acceleration in Pictures
Constant Velocity
Constant Acceleration
Average Acceleration
Acceleration is the rate of change of the velocity.
Dimensions are L/t2
SI units are m/s²
Example – Constant acceleration a=1 m/s2Time (s) Velocity (m/s)
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10
11 11
12 12
13 13
14 14
15 15
16 16
17 17
18 18
19 19
20 20
0
5
10
15
20
25
0 5 10 15 20 25
time (s)v
(m/s
)
atvv
vvatvtva
f
initialf
0
Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as t approaches 0
2
20lim x x
x t
v dv d xat dt dt
Calculus
2
2
dtxd
dtdva
dtdxv
Instantaneous Acceleration -- graph The slope of the
velocity vs. time graph is the acceleration
The green line represents the instantaneous acceleration
The blue line is the average acceleration
Acceleration and Velocity, 1
When an object’s velocity and acceleration are in the same direction, the object is speeding up
When an object’s velocity and acceleration are in the opposite direction, the object is slowing down
Example
Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same
length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration
Some math. Assume constant acceleration
atvv
atvv
attadtaadtvvdv
adtdv
adtdv
f
tv
v
t
f
f
0
0
000 )0(
constant
0
More …
200
200
000
0
0
0
2121
0
attvxx
attvxx
tdtadtvdx
atdtdtvdx
atvdtdx
atvv
ttx
x
One more calculus trick: the the chain rulechain rule
axvv
vvxxa
vdvdxa
vdvadxdxdvv
dtdx
dxdv
dtdva
f
f
v
v
x
x
f
2
0for x
)(21)(
20
2
0
20
20
00
Kinematic Equations -- summary from the book.
Kinematic Equations
The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration
You may need to use two of the equations to solve one problem
Many times there is more than one way to solve a problem
Kinematic Equations, specific
For constant a, Can determine an object’s velocity at any time t
when we know its initial velocity and its acceleration
Does not give any information about displacement
Kinematic Equations, specific
For constant acceleration,
The average velocity can be expressed as the arithmetic mean of the initial and final velocities
2xi xf
x
v vv
Kinematic Equations, specific
For constant acceleration,
Gives final position in terms of velocity and acceleration
Doesn’t tell you about final velocity
212f i xi xx x v t a t
Graphical Look at Motion – displacement – time curve The slope of the curve
is the velocity The curved line
indicates the velocity is changing Therefore, there is an
acceleration
Graphical Look at Motion – velocity – time curve The slope gives the
acceleration The straight line
indicates a constant acceleration
The zero slope indicates a constant acceleration
Graphical Look at Motion – acceleration – time curve
Let’s get real and throw something out of a building and watch it fall.This is called free fall.
Freely Falling Objects
A freely falling object is any object moving freely under the influence of gravity alone.
It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward
Acceleration of Freely Falling Object The acceleration of an object in free fall is directed
downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80
m/s2
g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the Earth’s surface
In the English/American system, g=32 ft/sec2.
Acceleration of Free Fall/Rise, cont. We will neglect air resistance Free fall motion is constantly accelerated
motion in one dimension Use the kinematic equations to solve
problems.
Free Fall ExampleLet’s drop an object from the topof this building by simply releasing it.
How long will it take to get to the ground?
How fast will it be going when it gets to theground?
What color is the object?
Issues –
Where is the origin?IS y the same thing as x?? How can that be???