phys 1443 – section 001 lecture #3 wednesday, january 26, 2011 dr. jaehoon yu one dimensional...
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PHYS 1443 – Section 001Lecture #3
Wednesday, January 26, 2011Dr. Jaehoon Yu
• One Dimensional Motion• Displacement• Average Speed and Velocity• Instantaneous Speed and Velocity• Acceleration
• 1D Motion under constant acceleration• Free Fall
Wednesday, Jan. 26, 2011 2PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Announcements• Some homework tips
– 57 out of 59 registered! • Very important! Please register ASAP
– When inputting answers to the Quest homework system• Unless the problem explicitly asks for significant figures, input as
many digits as you can• The Quest is dumb. So it does not know about anything other
than numbers• 49 out of 59 subscribed to e-mail list
– 3 point extra credit if done by today, Jan. 26• The first term exam is to be on Monday, Feb. 7
– Will cover CH1 – what we finish on Wednesday, Feb. 3 + Appendices A and B
– Mixture of multiple choices and free response problems
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Special Problems for Extra Credit• Derive the quadratic equation for Bx2-Cx+A=0
5 points• Derive the kinematic equation
from first principles and the known kinematic equations 10 points
• You must show your work in detail to obtain full credit
• Due at the start of the class, Monday, Feb. 7
ifxf xxavv xxi 222
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Displacement, Velocity and SpeedOne dimensional displacement is defined as:
ixxx f
The average velocity is defined as: xv
The average speed is defined as: Total Distance Traveled
Total Elapsed Timev
f
f
i
i
x x
t t
x
t
Displacement per unit time in the period throughout the motion
Displacement is the difference between initial and final potions of the motion and is a vector quantity. How is this different than distance?
Displacement
Elapsed Time
Unit? m/s
Unit? m
Unit? m/s
A vector quantity
A scalar quantity
A vector quantity
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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How much is the elapsed time? t= t 0- t
1x 2x
What is the displacement? =x 2x 1- x
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Displacement, Velocity and SpeedOne dimensional displacement is defined as:
ixxx f
The average velocity is defined as: xv
The average speed is defined as: Total Distance Traveled
Total Elapsed Timev
Can someone tell me what the difference between speed and velocity is?
f
f
i
i
x x
t t
x
t
Displacement per unit time in the period throughout the motion
Displacement is the difference between initial and final potions of the motion and is a vector quantity. How is this different than distance?
Displacement
Elapsed Time
Unit? m/s
Unit? m
Unit? m/s
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Difference between Speed and Velocity• Let’s take a simple one dimensional translation
that has many steps:Let’s call this line as X-axis
Let’s have a couple of motions in a total time interval of 20 sec.
+10m +15m
-15m-5m -10m
+5m
Total Displacement: x
Total Distance Traveled: D
Average Velocity: f
f
ix
i
x xv
t t
Average Speed: Total Distance Traveled
Total Time Intervalv
x
t
0
20 0( / )m s
≡xf −xi i ix x 0( )m
10 15 5 15 10 5 60( )m
60
20 3( / )m s
PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
Wednesday, Jan. 26, 2011 88
Example 2.1
2 1f ix x x x x • Displacement:
• Average Velocity: f
f
ix
i
x xv
t t
• Average Speed:
Total Distance Traveled
Total Time Intervalv
The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00-s time interval, the runner’s position changes from x1=50.0m to x2=30.5 m, as shown in the figure. What was the runner’s average velocity? What was the average speed?
30.5 50.0 19.5( )m
2 1
2 1
x x x
t t t
19.5
6.50( / )3.00
m s
50.0 30.5 19.56.50( / )
3.00 3.00m s
PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Instantaneous Velocity and Speed• Can average quantities tell you the detailed story of
the whole motion?
xx
vt
Δt 0lim
*Magnitude of Vectors are Expressed in absolute values
•Instantaneous speed is the size (magnitude) of the velocity vector:
xx
vt
Δt 0lim
• Instantaneous velocity is defined as:– What does this mean?
• Displacement in an infinitesimal time interval• Average velocity over a very, very short amount of time
NO!!
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Instantaneous Velocity
Time
Average Velocity
Instantaneous Velocity
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Position vs Time Plot
timet1 t2 t3t=0
Position
x=0
x11 2 3
1. Running at a constant velocity (go from x=0 to x=x1 in t1, Displacement is + x1 in t1 time interval)
2. Velocity is 0 (go from x1 to x1 no matter how much time changes)3. Running at a constant velocity but in the reverse direction as 1. (go
from x1 to x=0 in t3-t2 time interval, Displacement is - x1 in t3-t2 time interval)
It is helpful to understand motions to draw them on position vs time plots.
Does this motion physically make sense?
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Example 2.3
2x 1x
xv
x
(a) Determine the displacement of the engine during the interval from t1=3.00s to t2=5.00s.
Displacement is, therefore:
A jet engine moves along a track. Its position as a function of time is given by the equation x=At2+B where A=2.10m/s2 and B=2.80m.
(b) Determine the average velocity during this time interval.
1 3.00tx 22.10 (3.00) 2.80 21.7m 2 5.00tx 22.10 (5.00) 2.80 55.3m
2 1x x 55.3 21.7 33.6 m
x
t
33.6
5.00 3.00
33.6
16.8 /2.00
m s
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Example 2.3 cont’d
1 nn nCtCtdt
d
5.00xv t s
xv At2
Calculus formula for derivative
The derivative of the engine’s equation of motion is
(c) Determine the instantaneous velocity at t=t2=5.00s.
and 0Cdt
d
The instantaneous velocity at t=5.00s is
0lim
t
x
t
dx
dt 2d
At Bdt
2 5.00A 2.10 10.0 21.0 /m s
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Displacement, Velocity and Speed
dt
dx
t
xvx
lim0Δt
Displacement ixxx f
Average velocityt
x
tt
xxv
i
ix
f
f
Average speed Spent Time Total
Traveled Distance Totalv
Instantaneous velocity
Instantaneous speed dt
dx
t
xvx
lim0Δt
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Acceleration
• In calculus terms: The slope (derivative) of the velocity vector with respect to time or the change of slopes of position as a function of time
xa xv analogous to
xa dt
dx
t
xvx
lim0Δt
analogous to
Change of velocity in time (what kind of quantity is this?)•Definition of the average acceleration:
•Definition of the instantaneous acceleration:
xf
f
xi
i
v v
t t
xv
t
f
f
i
i
x x
t t
x
t
xv
t
Δt 0lim
xdv
dt d dx
dt dt
2
2
d x
dt
Vector
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Acceleration vs Time Plot
What does this plot tell you?Yes, you are right!!The acceleration of this motion is a constant!!
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Example 2.4
xa )/(2.40.5
21
0.5
021 2sm
A car accelerates along a straight road from rest to 75km/h in 5.0s.
What is the magnitude of its average acceleration?
xfv
xiv
)/(105.4
1000
36002.4 242
hkm
0 /m s
75000
3600
m
s21 /m s
xf xi
f i
v v
t t
xv
t
Wednesday, Jan. 26, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu
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Few Confusing Things on Acceleration• When an object is moving in a constant velocity (v=v0), there is
no acceleration (a=0)– Is there any acceleration when an object is not moving?
• When an object is moving faster as time goes on, (v=v(t) ), acceleration is positive (a>0).– Incorrect, since the object might be moving in negative direction initially
• When an object is moving slower as time goes on, (v=v(t) ), acceleration is negative (a<0)– Incorrect, since the object might be moving in negative direction initially
• In all cases, velocity is positive, unless the direction of the movement changes.– Incorrect, since the object might be moving in negative direction initially
• Is there acceleration if an object moves in a constant speed but changes direction? The answer is YES!!