chapter 2: motion in one dimension how to solve any physics problem: 1. diagram or draw a picture 2....
TRANSCRIPT
Chapter 2:Motion in One Dimension
How to solve any physics problem:
1. Diagram or draw a picture2. Units and variables labeled3. Formulas written and ready to use4. Algebra shown with numbers and units5. Solution boxed with correct units and sig. figs.
There are three ways to describe motion.
1. Displacement
2. Velocity
3. Acceleration
Displacement and Velocity
First off, to describe motion, we must specify position.
But to specify position, we need to relate it to another known location
You want to be here
You are here
From this concept, we can create a number line.
For motion in one dimension, it is convenient to use the x-axis.
0
+
0
-
0
Displacement
Displacement of an object - the change of position of the object
NOTE: displacement is not always equal to the distance traveled.
0
Displacement (x)
x = xf – xi Where x = displacement
xf = final position xi = initial position
Example
What is the displacement if a car moves from an initial position of 10 m to a final position of 80 m?
x =80 m – 10 m = 70 m
ExampleWhat is the displacement if a car moves from an initial position of 80 m to a final position of 20 m?
x =20 m – 80 m = -60 m
Note: the negative sign indicates the direction of displacement.
Distance and DisplacementDistance and Displacement
1. Your displacement is 0, distance traveled is 8 mi
2. Your displacement is 8 mi, distance traveled is 0
3. Your displacement is 0, distance traveled is 0
4. Your displacement is 8 mi, distance traveled is 8 mi
Every morning you drive 4 miles from home to NPHS and then come back home at night using the same route. What is the best statement describing your daily trip? (Assume you do not move far while at NPHS)
VELOCITY – change in displacement over time. Average velocity :
if
ifavg tt
xx
t
xv
)( avgv
Units for velocity
•The SI unit for velocity is m/s; however, a velocity can be given in other units such as:
•Km/sec or km/hr or mi/hr
•When calculating the velocity, be sure all your units match. If they don’t, you need to convert.
Example: It takes 6.0 hours to drive to San Francisco from Newbury Park if driving at 65 mi/hr. How far is the trip?
Newbury Park
San Fransiscot = 6.0 hrs
v = 65 mi/hrs
x = ?
t
xv
vtx
hrshr
mi0.665
milesx 390
Example: The United States is 4,300 km wide. How long (in hours) would it take to drive across the country if someone were to drive at a steady 22 m/s the whole way?
t = ?
v = 22 m/s
x = 4,300 km
t
xv
v
xt
sm
m
/22
000,300,4
sec195455t
km
m
1
1000 m000,300,4
min60
1
sec60
min1 hr hrs54
Speed vs. Velocity
Speed is a SCALAR meaning it is only a number, direction doesn’t matter. Speed = (total distance)/(time)
Velocity is a VECTOR meaning it has a number (magnitude) and direction. Direction matters. Velocity = (displacement)/(time) http://www.youtube.com/watch?v=GKQ
dkS0qG3g
Acceleration
Acceleration – rate of change in velocity over time.
if
ifavg tt
vv
t
va
onaccelerati average is where avga
Units of Acceleration
a
avg
v t
m / ss
ms
1s
ms 2
Example: A car is initially coasting up hill at 8.5 m/s. 4.75 seconds later, the car is now rolling backwards at 2.6 m/s down the same hill. What is the car’s acceleration during this time period?
+ Vo - V f
t = 4.75 sec
Vf = - 2.6 m/s
Vo = 8.5 m/s
t
vva of
75.4
5.86.2
s
sm
75.4
/1.11
2/3.2 sma
Another way to describe motion is to graph displacement and velocity as a function of time it
GLX Time!Plug motion sensor into one of the ports at the top (line the groove up)
A position vs time graph should automatically open
The “Play” button lets you start and stop data taking.
There is no need to erase your graph. Just hit “Play” again to take more data and overwrite your old graph.
Posi
tion
(m
)
Time (s)
(1)Create this graph on your GLX using your motion sensor.
(2) Describe your motion in your notes.
In this case, displacement stays constant with time…there is no movement.
Posi
tion
(m
)
Time (sec)
(1)Create this graph on your GLX using your motion sensor.
(2) Describe your motion in your notes.
Here the object has a low constant velocity in the beginning, and then it changes to a higher constant velocity.
Posi
tion
(m
)
Time (s)
(1)Create this graph on your GLX using your motion sensor.
(2) Describe your motion in your notes.
This time the object travels away from the starting point at constant velocity, stops for some amount of time and then travels back to its starting point at constant velocity.
Posi
tion
(m
)
Time (s)
Velocity is not constant. Velocity (slope) is increasing at a constant rate.
(1)Create this graph on your GLX using your motion sensor.
(2) Describe your motion in your notes.
Good luck on this one!
SPEED vs. VELOCITY Both describe how fast the position is changing with
respect to time.
Speed is a SCALAR quantity. It indicates an amount (magnitude), but not direction.
Velocity is a VECTOR quantity. It indicates both an amount (magnitude) and a direction.
SLOPE REVIEW
12
12xx
yy
run
risemslope
SLOPE REVIEW
SLOPE REVIEW
SLOPE REVIEW
SLOPE REVIEW
Posi
tion
(m
)
Time (s)
Rise
Run
Run
Riseslope
t
x v
Velocity is represented by the SLOPE of the curve on a displacement vs. time graph.
In this class, a positive (+) slope indicates a forward direction and a negative (-) slope indicates a backwards direction (return).
VELOCITY
Use the position graph to answer the following:
b. What is the object’s velocity from 15 – 25 seconds?
a. What is the object’s velocity from 10 – 15 seconds?
c. What is the object’s velocity from 0 – 40 seconds?
0 m/s (object is at rest)
run
riseslopev
ss
mm
1525
6020
= -4 m/s
run
riseslopev
ss
mm
040
040
= -1 m/s
Posi
tion
(m
)
Time (s)
Consider this trip…
Posi
tion
(m
)
Time (s)
INSTANTANEOUS VELOCITY
How fast is the car going at this instant in time?
Posi
tion
(m
)
Time (s)
INSTANTANEOUS VELOCITY
The slope of a curve at a given point, is equal to the slope of a tangent line at that point.
Posi
tion
(m
)
Time (s)
INSTANTANEOUS VELOCITY
The slope of this line represents instantaneous velocity at the indicated point.
Describing MotionDescribing Motion
1. Speeds up all the time
2. Slows down all the time
3. Speeds up part of the time
4. Slows down part of the time
5. Speeds up & then slows down
6. Slows down & then speeds up
Hint: What is the meaning of the slope of the
x(t) graph?
The x(t) graph describes a 1-D motion of a train. What must be true about this motion?
Position vs. Time Position vs. Time GraphsGraphs
The x(t) graph displays motions of two trains A and B on parallel tracks. Which statement is true?
1. At t1 both trains have the same velocity
2. At t1 both trains have the same speed
3. At t1 both trains have the same acceleration
4. Both trains have the same velocity sometime before t1
5. The trains never have the same velocity
B
t
x A
t1
End Position Graphs
Motion can be described with a velocity vs. time graph.
Time (s)
Vel
ocity
(m
/s)
constant velocity consta
nt acc
eleration
Velocity vs. Time 1
Time (s)
Vel
ocity
(m
/s)
0
+
-
+ velocity
0 velocity
- velocity
Velocity vs. Time 2
Vel
ocity
(m
/s)
Time (s)
Rise = v
Run = t
t
vSlope aonaccelerati
The Slope of Velocity vs. Time Graphs
Use the velocity graph to answer the following:
b. What is the object’s acceleration from 4 – 7 seconds?
a. What is the object’s velocity from 4 – 7 seconds?
c. What is the object’s acceleration from 2 – 4 seconds?
3 m/s
run
riseslopea
ss
smsm
24
/1/3
= 1 m/s2
0 m/s2 (constant velocity)
Consider a trip…
A train travels at 5 miles per hour for 1 hour. What is its displacement after 1 hour?
t
xv
tvx
hr) mi/hr)(1 5(xmi 5x
Displacement can also be determined by finding the area under the curve of a velocity vs. time graph.
5 mi/hr
1 hour
Vel
ocity
(m
/s)
Time (hrs)
Using a graph
5 mi/hr
1 hour
Area = l x w = 1 hour x 5 mi/hr = 5 miles.
Find the “area under the curve” (AUC).V
eloc
ity (
m/s
)
Time (s)
Displacement vs. Time graph:Slope = velocity
Velocity vs. Time graph:
Slope = acceleration
AUC = displacement
Graphical analysis summary
A blue car moving at a constant speed of 10 m/s passes a red car that is at rest. This occurs at a stoplight the moment that the light turns green.
The red car accelerates from rest at 4 m/s/s for three seconds and then maintains a constant speed for 9 s. The blue car maintains a constant speed of 10 m/s for the entire 12 seconds.
Draw a “v vs. t” graph for:
The answer
QUICK QUIZ 2.3Parts (a), (b), and (c) of the figure below represent three graphs of the velocities of different objects moving in straight-line paths as functions of time. The possible accelerations of each object as functions of time are shown in parts (d), (e), and (f). Match each velocity-time graph with the acceleration-time graph that best describes the motion.
One way to look at the motion of an object is by using a motion map.
Imagine a toy car traveling along a piece of paper and dropping a dot of ink at a given time interval (say 1 drop every second). It could produce a trail that looks like this:
distance
Same time intervalBetween dots
Position and directionof object at a given
instant in timeTime starts
at zero
distance
Same time intervalBetween dots
Position and directionof object at a given
instant in timeTime starts
at zero
If the car produced the following motion map, how long did it take the car to travel the length of the paper?
5 sec1 sec 2 sec 3 sec 4 sec 6 sec 8 sec7 sec
8 seconds total
distance
Same time intervalBetween dots
Position and directionof object at a given
instant in timeTime starts
at zero
Describe the motion of the car? Stopped, constant velocity, accelerating or
decelerating?
How do you know?
Graphing Examples
a. Draw the motion map for the following:
•Object accelerates for 3 seconds.
•Then travels at a constant velocity for 2 seconds
•Then decelerates for 3 seconds
•Stops for 2 seconds
•Then returns to the start in 4 seconds at a constant velocity.
b. Sketch the velocity graph for the above motion.
time
velo
city
v
vo
ttime
Velocity with Constant Acceleration
time t
vf
vovelo
city Rise = v
= vf - vo
Run = t = (t – 0) = t
slopevt
v
f v
o
ta
vfv
oat
(p. 35)
time t
vf
vivelo
city
Find the AUC…
Break the area into two parts…
x = vit + ½ t(vf-vi)
Area = (l x w) + (½b x h)
time t
vf
vivelo
city
Find the AUC…
AUC = x = vot + ½ t(vf-vo)
x = ½ vot + ½ vft simplification
x = vot + ½ vft – ½ vot distribution
x = ½ (vo+vf)t p.(35)
1) vf = vo + at
x = ½ [vo + (at +vo)]t substitution
and 2) x = ½ (vo + vf)t
x = ½ (2vo + at)t combining terms
x = vot + ½ at2 (p. 36)
We now know…
By substitution of equation #1 into #2
Solve Eq #1 for t & substitute into Eq #2…
1) vf = vo + at
tvf v
o
a
2) x = ½ (vo + vf)t
(p. 36)
x (vov
f)
2(v
f v
o)
a
x vf
2 vo
2
2av
f
2 vo
2 2ax
1. a = acceleration
2s
m
time
velocity
2. x = displacement m
3. vf = final velocity s
m
4. vi = initial velocity
5. t = time sec
s
m
5 Parameters of Motion
To solve a constant acceleration problem, you must know, or be able to find, three of the five parameters.
Then use the following equations to solve for the other two:
vf = vo + at
x = ½(vo + vf)t
x = vot + ½a(t)2
vf2 = vo
2 + 2ax
Example…Example: A jet plane lands with a velocity of 100 m/sec and can slow down (-acceleration) at a maximum rate of –5.0 m/s2. Find (a) the time required for the plane to come to rest, and (b) the minimum size of the runway.
a) vi = +100m/svf= 0 m/sa = -5.0 m/s2
vf = vi + at
0m/s = 100m/s + (-5m/s2)t
t = 20s
vf2 = vi
2 + 2ax
0m/s = (100m/s)2 + 2(- 5m/s2) x
x = 1000 m
b) Solve for x
Example: A train is traveling down a straight track at 20 m/sec when the engineer applies the brakes, resulting in an acceleration of –1m/sec2 as long as the train is in motion. How far does the train travel in the first 6 seconds after the breaks are applied?
vi = 20 m/s
a = -1 m/sec2
t = 6 sec
x = ?
x = vit + ½ at2
x = (20m/s)(6s) + ½ (-1m/s2)(6s)2
x = 120m – 18m = 102 m
x = 100 m (sig. figs!)
Example: A racing car starting from rest accelerates at a rate of 5.00 m/s2. What is the velocity of the car after it has traveled 100. ft?
vi = 0 m/s
a = 5.0 m/sec2
x = 100 ft= 30.5 m
vf = ?
vf2 = vi
2 + 2ax
vf2 = 0 + 2(5m/s2)(30.5m)
vf = 17.5 m/s
22f /sm 305v