Mechanics 105 Motion diagrams, position-time graphs, etc. Average and instantaneous velocity...
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Transcript of Mechanics 105 Motion diagrams, position-time graphs, etc. Average and instantaneous velocity...
Mechanics 105
Motion diagrams, position-time graphs, etc.
Average and instantaneous velocity
Acceleration
Particle under constant acceleration
Freefall
Motion in one dimension (chapter two)
Mechanics 105
Motion diagrams
Motion diagram:
Mechanics 105
Position vs. time
x vs. t
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time (seconds)
po
sit
ion
(m
)
Mechanics 105
Velocity vs. time
0
2
4
6
8
10
0 5 10 15 20 25 30
time (seconds)
po
siti
on
(m
)
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20 25 30
time (seconds)
velo
city
(m
/s)
Mechanics 105
Position vs. time
Demo - ~constant velocity
Conceptests
http://webphysics.davidson.edu/physletprob/ch7_in_class/in_class7_1/mechanics7_1_1.html
http://webphysics.davidson.edu/physletprob/ch8_problems/ch8_1_kinematics/default.html
Mechanics 105
only depends on beginning and end points and time interval vector (scalar in 1-d only) total displacement = integral of v(t)dt
examples:
A person runs 1 km in 5 minutes, then walks another 2 km in 20 minutes. What is their average velocity over the entire 3 km?
Using the velocity vs. time graph shown earlier, find the average velocity and the total displacement
Average velocity
f
i
f
i
x
x
x
x
if dtdt
tdxdttvxxx
)()(
Mechanics 105
Instantaneous velocity
limit of average velocity as interval goes to zero tangent to x(t) - derivative of x(t) vs. t examples
A ball rolling down a slope has a position described by the equation
What is the equation describing the instantaneous velocity? ConcepTests
22 )/5.0()/1.0(50.0)( tsmtsmmtx
Mechanics 105
Particle under constant velocity
relation between initial and final displacement comes from the definition of average velocity
average equals instantaneous
tvxxvt
xif
Mechanics 105
Acceleration
average
instantaneous
t
vva if
2
2
0lim)(
dt
xd
dt
dv
t
vta
t
Mechanics 105
Particle moving under constant acceleration
tavvt
vva if
if
Displacement of particle = integral of velocity as a function of time
From definition of average acceleration:
t)(tattvxx
tatvdtatvdttvx
iif
t
t
ii
t
t
f
i
f
i
2
1
2
1)()(
2
2
Mechanics 105
Particle moving under constant acceleration
tvvxtt
vvtvxattvxx
t
vvaatvv
ifiif
iiiif
ifif
)(2
1)(
2
1
2
1
)(
22
Or (removing t)
Combining previous two equations (removing a)
)(2
)(
2
1)(
2
1
)(
22
2
22
ifif
ififiiiif
ifif
xxavv
a
vva
a
vvvxattvxx
a
vvtatvv
Mechanics 105
Particle moving under constant acceleration
Wide range of applications
Zero acceleration:
Free fall:
acceleration due to gravity = 9.80 m/s2 downward (be careful about sign)
If we define up to be the direction of the positive y axis, the equations of motion for a particle in free fall are:
tvxx iif
gtvvgttvyy iii and 2
1 2
Note that the velocity can be positive or negative
Mechanics 105
Particle moving under constant acceleration
ConcepTests
Demo – cart
More physlets
http://webphysics.davidson.edu/physletprob/ch8_problems/ch8_1_kinematics/default.html
Examples
Mechanics 105
Example - braking distance (from Giancoli, 2-10)
Estimate minimum stopping distance for a car traveling at 60 mph
1) Maximum (negative) acceleration? 5~8 m/s2 (dry road, good tires)
2) Typcial human response time? 0.3 ~ 1.0 sec (sober)
part 1 – distance before brakes applied:
part 2 – distance until car stops:
smmile
mhour
hour
miles/8.26
1
1610
sec3600
160
msm
smsmsm
a
vvxx 8.97
)/00.8(
)/8.26()/000.0(/04.8
2 2
2220
2
0
msmtvx 04.8sec)300.0)(/8.26(0
Mechanics 105
Example – Air bags (Giancoli 2-11)
If, instead of braking, the car in the previous example hit a tree, estiamte how fast the air bags need to inflate to do any good.
estimate a stopping distance ~ 1.00m
initial velocity = 26.8 m/s
final velocity = 0.00 m/s
first solve for a:
then find t
222
/35900.2
)/8.26(
2sm
m
sm
x
va f
sec0747.0/359
)/8.26/00.0()(2
sm
smsm
a
vvt if
Mechanics 105
Problem solving
Choice of coordinate system can simplify problem
Be consistent with signs (direction of chosen axis)
Often problems involve two or more objects with some common variable (time, final displacement, etc.)