PHYS 1441 – Section 501
Lecture #2
Monday, June 7, 2004
Dr. Jaehoon Yu
•
Chapter two: Motion in one dimension
–
–
–
–
Velocity (Average and Instantaneous)
Acceleration (Average and instantaneous)
One dimensional motion at constant
acceleration
• Free Fall
Coordinate systems
Remember the quiz this Wednesday!!
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
1
Announcements
• Reading assignment #1: Read and follow through all sections
in appendix A by Wednesday, June 9
– A-1 through A-9
• There will be a quiz on this Wednesday, June 9, on these and
Chapter 1
• E-mail distribution list: 16 of you have registered
– Remember 5 (3) extra credit points if done by midnight tonight
(Wednesday).
• Homework: You are supposed to download the homework
assignment, solve it offline and input the answers back
online.
– 38 registered
– 25 submitted
– Must be submitted by 6pm Wednesday to get full, free credit.
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
2
Displacement, Velocity and Speed
One dimensional displacement is defined as:
x  xf  xi
Displacement is the difference between initial and final
potions of motion and is a vector quantity
Average velocity is defined as: vx  xf  xi  x
tf  ti
t
Displacement per unit time in the period throughout the motion
Average speed is defined as:
Total Distance Traveled
v
Total Time Interval
Can someone tell me what the difference between speed and velocity is?
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
3
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
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
20
4
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
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
5
Instantaneous Velocity and Speed
• Can average quantities tell you the detailed story of the
whole motion?
• 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 short amount of time
•Instantaneous speed is the size (magnitude) of the
velocity vector:
*Magnitude of Vectors
vx 
lim
Δt 0
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
x
t
are Expressed in
absolute values
6
Position vs Time Plot
Position
It is useful to understand motions to
draw them on position vs time plots.
x1
x=0
2
1
t=0
1.
2.
3.
Monday, June 7, 2004
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 1441-501, Summer 2004
Dr. Jaehoon Yu
7
Velocity vs Time Plot
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
8
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
Instantaneous speed
vx 
Monday, June 7, 2004
Δt 0
x
t
lim
Δt 0
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
x
t
9
Acceleration
Change of velocity in time (what kind of quantity is this?)
•Average acceleration:
vxf  vxi v
x  xi x
ax 


analogs to vx 
t  ti
t
t
tf  ti
f
x
f
•Instantaneous acceleration: Average acceleration
over a very short amount of time.
ax  lim
Δt 0
Monday, June 7, 2004
vx
t
analogs to
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
vx  lim
Δt 0
x
t
10
Acceleration vs Time Plot
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
11
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
Monday, June 7, 2004
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 1441-501, Summer 2004
Dr. Jaehoon Yu
12
Meanings of 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!!
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
13
One Dimensional Motion
• Let’s start with the simplest case: acceleration is a constant (a=a0)
• Using definitions of average acceleration and velocity, we can
derive equations of motion (description of motion, velocity and
position as a function of time)
vxf  vxi
vxf  vxi
a

(If tf=t and ti=0) x
a x  ax 
t
tf  ti
For constant acceleration, average
velocity is a simple numeric average
vx 
xf  xi
tf  ti
(If tf=t and ti=0)
Resulting Equation of
Motion becomes
Monday, June 7, 2004
vxf  vxi  axt
vxi  vxf 2vxi  axt
1

 vxi  axt
vx 
2
2
2
xf  xi
vx 
t
xf 
xf  xi  v t
x
1 2
xi  v xt  xi  vxit  axt
2
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
14
One Dimensional Motion cont’d
Average velocity
Since
 vxi  vxf
x f  xi  v xt  xi  
 2
vxi  vxf
vx 
2
vxf  vxi
ax 
t
Solving for t
vxf  xxi
t
ax
Substituting t in the
 vxf  vxi   vxf  vxi 
x f  xi  
  xi 

above equation,

Resulting in
Monday, June 7, 2004
2

ax

vxf2  vxi2
2a x
v  v  2ax  x f  xi 
2
xf
2
xi
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu

t

15
Kinematic Equations of Motion on a
Straight Line Under Constant Acceleration
vxf t   vxi  axt
Velocity as a function of time
1
1
xf  xi  v x t   vxf  vxi  t Displacement as a function
of velocities and time
2
2
1 2
xf  xi  vxit  axt
2
vxf  vxi  2axxf  xi 
2
2
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!!
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
16
How do we solve a problem using a kinematic
formula for constant acceleration?
• Identify what information is given in the problem.
–
–
–
–
Initial and final velocity?
Acceleration?
Distance?
Time?
• Identify what the problem wants.
• Identify which formula is appropriate and easiest to solve for
what the problem wants.
– Frequently multiple formulae can give you the answer for the quantity
you are looking for.  Do not just use any formula but use the one
that can be easiest to solve.
• Solve the equation for the quantity wanted
Monday, June 7, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
17
Example 2.10
Suppose you want to design an air-bag system that can protect the driver in a
head-on collision at a speed 100km/s (~60miles/hr). Estimate how fast the air-bag
must inflate to effectively protect the driver. Assume the car crumples upon impact
over a distance of about 1m. How does the use of a seat belt help the driver?
How long does it take for the car to come to a full stop?
As long as it takes for it to crumple.
100000m
vxi  100km / h 
 28m / s
The initial speed of the car is
3600 s
v xf  0m / s and xf  xi  1m
We also know that
Using the kinematic formula
The acceleration is
ax 
vxf
2
vxf 2  vxi 2 0   28m / s  390m / s 2


2  xf  xi 
2 1m
2
Thus the time for air-bag to deploy is
Monday, June 7, 2004
2
v
xi
 2ax  xf  xi 

t
vxf  vxi
a
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
0  28m / s


 0.07s
2
390m / s
18