PHYS 1443 – Section 001
Lecture #3
Wednesday, January 26, 2011
Dr. Jaehoon Yu
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One Dimensional Motion
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Displacement
Average Speed and Velocity
Instantaneous Speed and Velocity
Acceleration
1D Motion under constant acceleration
Free Fall
Announcements
• Some homework tips
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• Very important! Please register ASAP
– When inputting answers to the Quest homework system
• Unless the problem explicitly asks for significant figures, input as
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– 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
2
Special Problems for Extra Credit
• Derive the quadratic equation for Bx2-Cx+A=0
 5 points
• Derive the kinematic equation v  vxi  2axx  x 
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
2
xf
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
2
f
i
3
Displacement, Velocity and Speed
One dimensional displacement is defined as:
x  xf  xi
A vector quantity
Displacement is the difference between initial and final potions of the
motion and is a vector quantity. How is this different than distance?
Unit?
m
xf  xi x Displacement


t Elapsed Time
tf  ti
Unit? m/s
A vector quantity
Displacement per unit time in the period throughout the motion
The average velocity is defined as: vx 
Total Distance Traveled
The average speed is defined as: v 
Total Elapsed Time
Unit?
m/s
Wednesday, Jan. 26, 2011
A scalar quantity
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
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x2
x1
What is the displacement?
How much is the elapsed time?
Wednesday, Jan. 26, 2011
x= x 2- x1
t= t - t 0
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
5
Displacement, Velocity and Speed
One dimensional displacement is defined as:
x  xf  xi
Displacement is the difference between initial and final potions of the
motion and is a vector quantity. How is this different than distance?
Unit?
m
xf  xi x Displacement


t Elapsed Time
tf  ti
Unit? m/s
Displacement per unit time in the period throughout the motion
The average velocity is defined as: vx 
Total Distance Traveled
The average speed is defined as: v 
Total Elapsed Time
Unit?
m/s
Can someone tell me what the difference between speed and velocity is?
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
6
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.
+15m
+10m
-5m
-10m
+5m
-15m
Total Displacement: x  x  xi  xi  xi  0(m)
f
Average Velocity:
vx 
xf  xi x 0
 0(m / s)


tf  ti
t 20
Total Distance Traveled: D 10  15  5  15  10  5  60(m)
Average Speed: v  Total Distance Traveled  60  3(m / s)
Total Time Interval
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
20
7
Example 2.1
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?
• Displacement:
x  x  xi  x2  x1  30.5  50.0  19.5(m)
• Average Velocity:
f
vx 
xf  xi x2  x1 x 19.5

 6.50(m / s )


t2  t1
t
3.00
tf  ti
• Average Speed:
Total Distance Traveled
v
Total Time Interval
50.0  30.5 19.5


 6.50(m / s )
3.00
3.00
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? NO!!
• Instantaneous velocity is defined as:
x
vx  lim
– What does this mean?
Δt 0
t
• Displacement in an infinitesimal time interval
• Average velocity over a very, very short amount of time
•Instantaneous speed is the size (magnitude) of the
velocity vector:
*Magnitude of Vectors
vx 
lim
Δt 0
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
x
t
are Expressed in
absolute values
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Instantaneous Velocity
Instantaneous Velocity
Average Velocity
Time
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
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Position vs Time Plot
Position
It is helpful to understand motions to
draw them on position vs time plots.
x1
x=0
2
1
t=0
1.
2.
3.
Wednesday, Jan. 26, 2011
t1
3
t2
t3
time
Running at a constant velocity (go from x=0 to x=x1 in t1,
Displacement is + x1 in t1 time interval)
Velocity is 0 (go from x1 to x1 no matter how much time changes)
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)
Does this motion physically make sense?
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
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Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
12
Example 2.3
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.
(a) Determine the displacement of the engine during the interval from t1=3.00s to t2=5.00s.
x1  xt1 3.00  2.10  (3.00)2  2.80  21.7m
Displacement is, therefore:
x2  xt 5.00  2.10  (5.00)2  2.80  55.3m
2
x x2  x1  55.3  21.7  33.6  m
(b) Determine the average velocity
x

vx 
during this time interval.
t
Wednesday, Jan. 26, 2011
33.6
33.6

 16.8  m / s 
5.00  3.00 2.00
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
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Example 2.3 cont’d
(c) Determine the instantaneous velocity at t=t2=5.00s.
 
d
Ct n  nCt n 1 and
Calculus formula for derivative
dt
The derivative of the engine’s
equation of motion is
vx  lim
t 0
d
C   0
dt
x dx d
   At 2  B   2 At
t dt dt
The instantaneous v t  5.00s 
2 A 5.00  2.10 10.0  21.0  m / s 

x
velocity at t=5.00s is
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
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Displacement, Velocity and Speed
Displacement
x  xf  xi
Average velocity
xf  xi x
vx 

tf  ti t
Average speed
Total Distance Traveled
v
Total Time Spent
Instantaneous velocity
vx  lim
x dx

t dt
Instantaneous speed
vx 
lim
x dx

t
dt
Wednesday, Jan. 26, 2011
Δt 0
Δt 0
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
15
Acceleration
Change of velocity in time (what kind of quantity is this?) Vector
•Definition of the average acceleration:
vxf  vxi vx

ax 
t
tf  ti
analogous to
xf  xi x

vx 
tf  ti t
•Definition of the instantaneous acceleration:
ax 
lim
Δt 0
vx dvx d  dx  d 2 x
  

dt dt  dt  dt 2
t
analogous to
vx  lim
Δt 0
x dx

t dt
• 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
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
A car accelerates along a straight road from rest to 75km/h in 5.0s.
What is the magnitude of its average acceleration?
vxi  0 m / s
v xf 
75000m
 21 m / s
3600 s
Wednesday, Jan. 26, 2011
v x
21  0 21



 4.2(m / s 2 )
ax 
t
5.0
5.0
t f  ti
vxf  vxi
4.2  3600

 5.4 10 4 (km / h 2 )
1000
2
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!!
Wednesday, Jan. 26, 2011
PHYS 1443-001, Spring 2011 Dr.
Jaehoon Yu
19