wednesday, sept. 15, 2010 phys 1441-002, fall 2010 dr. jaehoon yu 1 phys 1441 – section 002...
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Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
1
PHYS 1441 – Section 002Lecture #4
Wednesday, Sept. 15, 2010Dr. Jaehoon Yu
• One Dimensional Motion• Instantaneous Velocity and Speed• Acceleration• Motion under constant acceleration• One dimensional Kinematic Equations• How do we solve kinematic problems?• Falling motions
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Wednesday, Sept. 15, 2010
2PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
4
Reminder: Special Problems for Extra Credit• Derive the quadratic equation for yx2-zx+v=0
5 points• Derive the kinematic equation
from first principles and the known kinematic equations 10 points
• You must show your OWN work in detail to obtain the full credit• Must be in much more detail than in pg. 19 of this lecture note!!!
• Due Monday, Sept. 27
2 2
0 02v v a x x
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 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
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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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
Wednesday, Sept. 15, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s?
Average speed
Distance
Example Distance Run by a Jogger
Distance Elapsed time
Average speed Elapsed time
2.22 m s 5400 s 12000 m
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. To establish such a record, the driver makes two runs through the course, one in each direction to nullify wind effects. From the data, determine the average velocity for each run.
Example: The World’s Fastest Jet-Engine Car
vr
t
xr
1609 m339.5m s
4.740 s
vr
t
xr
1609 m342.7 m s
4.695 s
What is the speed?
What is the speed?
v vr
v vr
339.5m s
342.7 m s
342.7 m s
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
9
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 short amount of time
NO!!
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 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 useful to understand motions to draw them on position vs time plots.
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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Velocity vs Time Plot
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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Displacement, Velocity and Speed
xx
vt
Δt 0lim
Displacement ixxx f
Average velocityt
x
tt
xxv
i
ix
f
f
Average speed Spent Time Total
Traveled Distance Totalv
Instantaneous velocity
Instantaneous speed xx
vt
Δt 0lim
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
14
Acceleration
xa xv analogs to
xa xx
vt
Δt 0limanalogs to
Change of velocity in time (what kind of quantity is this?)
•Average acceleration:
•Instantaneous acceleration: Average acceleration over a very short amount of time.
xf
f
xi
i
v v
t t
xv
t
f
f
i
i
x x
t t
x
t
xv
t
Δt 0
lim
Vector!
Unit? m/s2Dimension? [LT-2]
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 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, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
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Example 2.3
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, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
17
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!!
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
18
One Dimensional Motion• Let’s focus on the simplest case: acceleration is a constant (a=a0)• Using the definitions of average acceleration and velocity, we can derive
equations of motion (description of motion, velocity and position as a function of time)
xa (If tf=t and ti=0) xfv
For constant acceleration, average velocity is a simple numeric average
xv
fx Resulting Equation of Motion becomes
xv
fx
xi xv a txa
xf
f
xi
i
v v
t t
xf xi
x
v va
t
2
xi xfv v
2
2
xi xv a t
1
2xi xv a t
f
f
i
i
x x
t t
f ixx x
vt
(If tf=t and ti=0) xix v t
xix v t 21
2i xi xx v t a t
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
19
One Dimensional Motion cont’d
fx
xa
Average velocity
t Since
fx
Substituting t in the above equation,
xv
2xfv
Solving for t
Resulting in
2
xi xfv vxix v t
2xi xf
i
v vx t
xf xiv v
t
xf xi
x
v x
a
2xf xi xf xi
ix
v v v vx
a
2 2
2xf xi
ix
v vx
a
2 2xi x f iv a x x
Wednesday, Sept. 15, 2010
PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu
20
Kinematic Equations of Motion on a Straight Line Under Constant Acceleration
tavtv xxixf
2
2
1tatvxx xxiif
1 1
2 2f xi xf xix x v t v v t
ifxf xxavv xxi 222
Velocity as a function of time
Displacement as a function of velocities and time
Displacement as a function of time, velocity, and acceleration
Velocity as a function of Displacement and acceleration
You may use different forms of Kinetic equations, depending on the information given to you for specific physical problems!!