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Kinematics CH 3 Describing motion
What is Motion?
any physical movement or change in position or place, relative to a reference point
Reference Point
Movement
Motion diagrams
At rest Moving at a constant speed
Speeding up Slowing down
A series of images of a moving object that records its
position after equal time intervals.
It represents the position, velocity and acceleration
of an object at several different times.
the particle model
What is the motion of the cart in this
diagram?
Slowing down (deceleration)
Change is position
is less/time interval
We can use motion diagrams to
represent 4 concepts in kinematics:
Uniform motion (constant speed)
At rest
Speeding up (acceleration)
Slowing down (deceleration)
Choose the coordinates
Establish where “0” is (the origin)
Establish directions where the values increase
Using a coordinate system
example - football
What origins and direction are used to determine 1st down?
Length of punt?
Position vector
Proportional to the distance of the object from the
origin and points from origin to the location of the
object at a particular time.
0
+
+
When would position be –
but the displacement +?
distance and displacement
Distance refers to how much ground an object
has covered during its motion.
X Axis
Y Axis
Distance = how far an object has moved.
Measured in meters, kilometers (cm or mm)
If each mark represents 10 cm, what is the distance between the girl and the ball? ______
displacement ≠ distance
Displacement is the object's change in
position.
X Axis
Y Axis If the girl walks to the red ball, then walks backwards to the bear, what distance has she traveled? ______
Displacement = the distance of a body's change in position from a starting point. Her final displacement is ______.
©2008 by W.H. Freeman and Company
The distinction between distance and displacement.
Displacement (blue line) is how far the object is from its starting point,
regardless of how it got there.
Distance traveled (dashed line) is measured along the actual path.
As any object moves from one position to another,
the length of the straight line drawn from its initial
position to the object’s final position is called
displacement.
Displacement doesn’t always tell you distance an
object moved.
Considering
displacement
∆x = xf - xi
Math and physics
Physicists use the tools of math to describe measured or
predicted relationships between physical quantities in a
situation. Equation = a compact statement
based on a model of the situation.
Shows how 2 or more variables are thought to be related.
Physics shorthand
∆ “difference” or “change in”
∑ “sum” or “total”
∆x, ∆y change in position
∆t time interval
Distance and Displacement
Scalar - magnitude only
Vector - magnitude AND direction
Distance is the length of the actual path taken by an object. Consider travel from point A to point B in diagram below:
Distance s is a scalar
quantity:
Contains magnitude only
and consists of a number
and a unit.
ex: 20 m, 40 mi/h, 10 gal
A
B s = 20 m
Displacement is the straight-line separation of
two points in a specified direction.
A vector quantity:
contains magnitude AND
direction.
A
B D = 12 m, 20o
q
ex: 12 m, 300; 8 km/h, N
In th e d ia gra m below, th e or igin , or in it ia l pos it ion (d o) is a t 0 .0 m . Th e
fin a l pos it ion (d 1) is a t 50 m . Th e d is ta n ce t ra veled from th e or igin is 50
m , bu t th e d is p la cem en t is 50 m to th e r igh t , or ca n be d ra wn s im ila r with
a green lin e – vec t or .
How are distance and displacement related to
motion?
Motion of an object is BOTH
a scalar quantity (time)
And
Vector quantity (displacement)
xi
initial position
xf
final position
Displacement: ∆x = xf – xi
displacement = distance and direction between
2 positions = change in position =
final position – initial position
• If displacement is positive, the object
moves to the right.
• If the displacement is negative, the object
moves to the left.
Defining the reference point and direction
The values of xi and xf are determined by their positions on
the axis.
While the choice of a reference point for the coordinate
system is arbitrary, once chosen, the same point must
be used throughout the problem.
Signs of Displacement
In ph ys ics , th e m ovem en t from th e or igin is th ou gh t of a s
pos it ive or n ega t ive. In ea ch ca s e, a n or igin , s ta r t in g p la ce,
or referen ce poin t n eeds to be es ta b lis h ed . Th en , it m u s t be
decided wh ich d irect ion s a re con s idered pos it ive a n d wh ich is
con s idered n ega t ive. On ce a grou p a grees on th a t , th en you
ca n deter m in e d is p la cem en t vectors . For exa m ple, if we
a s s u m e th a t u p is pos it ive, th en Mt. Elin or, wou ld h a ve a n
eleva t ion d is p la cem en t vector
of +2 ,400 ft , ba s ed off of th e or igin
of s ea level. On th e oth er h a n d ,
Dea th Va lley, Ca lifor n ia is below
s ea level by 120 ft , s o its
d is p la cem en t wou ld be -120 ft
com pa red to s ea level.
©2008 by W.H. Freeman and Company
Left:
Displacement is positive.
Right:
Displacement is negative.
The displacement is written:
The Signs of Displacement • Displacement is positive (+) or negative
(-) based on LOCATION.
2 m
-1 m
-2 m
The displacement is the
y-coordinate. Whether
motion is up or down,
+ or - is based on
LOCATION.
Examples:
The direction of motion does not matter!
• For motion along x or y axis, the displacement is determined by the x or y coordinate of its final position.
• Consider a car that travels 8 m, E then 12 m, W.
Net displacement “D” is from the
reference point to the final position:
What is the distance
traveled? 20 m !! 12 m,W
D
D = 4 m, W x
8 m,E
x = +8 x = -4
What is this
measuring?
• Would it be good to measure MPG (miles per gallon) in position, displacement or distance?
•Distance
•Displacement
(Not the speedometer sillies)
Displacement and Distance
• A person starts at the 5.o m mark. They walk
to the 12m mark.
– What is their distance travelled?
– What is their displacement?
• They leave the 12m mark and walk to the 1m
mark.
– What is their distance travelled?
– What is their displacement?
• What is the total distance travelled?
• What is the total displacement for the motion?
• What is the distance
travelled for the first three
seconds?
• When is the object not
moving?
• What is the final speed of
the object at t=3 seconds?
• Rank speed at t=0,1,3
• What is the average velocity
for the entire motion?
What about at angles?
• You drive 3 miles east and then 4 miles
north. What is your:
– Why are you not drawing a picture people?
– Distance travelled?
– Displacement?
Strategies in Solving Problems
1. Picture the problem (See it…)
2. What am I answering?
3. What is important? (variables)
4. What will get me there?
5. Solve it.
• This year, the most important thing in the course is getting stronger at the top four. It’s the hardest thing, but transfers to everything in life.
In this motion diagram the length of the arrow indicates
the change in position of the object, or its _____.
Check it:
a. Displacement
b. Magnitude
c. Position
d. Resultant
In this image, 7 cm is a _____.
a. Vector
b. speed
c. Scalar
d. interval
VELOCITY, SPEED,
AND
ACCELERATION
The Cheetah: A cat that is built for speed. Its strength and
agility allow it to sustain a top speed of over 100 km/h.
Such speeds can only be maintained for about ten seconds.
Definition of Speed • Speed is the distance traveled per unit of time
(a scalar quantity).
s = = d
t
20 m
4 s
s = 5 m/s
Not direction dependent!
A
B s = 20 m
Time t = 4 s
Speed (meters/second) = distance (in meters) time (sec)
s = d t
Speed equation
velocity
A quantity that measures how fast
an object moves from one point to
another in a certain direction.
Definition of Velocity • Velocity is the displacement per unit of
time. (A vector quantity.)
v = 3 m/s at 200 N of E
Direction required!
A
B d = 20 m
Time t = 4 s
12 m
4 s
Dv
t
D=12 m
20o
Velocity
Velocity is a measure of the speed of an object AND the direction it is moving in space. On the escalator, passengers are moving at the same constant speed, but they are moving in different directions. Velocity can change even if speed is remaining constant (you just change direction)
Velocity is defined as a vector quantity that tells the
ratio of the displacement change to the time
change,
or
how fast an object is going and in what direction.
Speed, on the other hand, is merely the magnitude
of the velocity, or how fast an object is moving.
speed and velocity are NOT the same
Velocity – the rate at which an object changes its
position (has direction)
Speed – is the magnitude of velocity (how fast an object is
moving)
In other words: the average Velocity depends
on total displacement
BUT
the average Speed = total distance traveled
time interval
1) Ave rage s pe e d – a vera ge of a ll you r s peeds over th e wh ole t r ip . For exa m ple, on a t r ip to Flor ida , (910 m iles ) th e t ra veler took 13 h ou rs to
get th ere. Th e a vera ge s peed wou ld be 70 m ph .
Th is does n ’t m ea n th a t th e ca r wa s goin g exa ct ly 70 m i/ h r th e en t ire
t im e. Som etim es th e ca r wa s goin g fa s ter, a n d oth ers s lower.
2) Con s t an t Spe e d - t ra velin g a t th e s a m e ra te for a lon g per iod of
t im e. Con s ta n t s peed is h a vin g th e cru is e con trol on in th e ca r. Th e ca r
m a in ta in s th e s a m e s peed th e en t ire t im e you a re clock in g it .
3) In s t an t an e ous s pe e d - ra te a t wh ich a n ob ject is t ra velin g a t a
cer ta in m om en t . Th is is you r s peedom eter in you r ca r. It tells you h ow
fa s t th e ca r is goin g a t th e t im e you look a t it .
Th ere a re th ree types of s peed :
©2008 by W.H. Freeman and Company
Average Speed & Velocity
Velocity includes directional information:
t
x
tt
xxv
12
12
Average velocity = change in position
change in time
= displacement
time interval
V avg = ∆x = xf – xi
∆t tf - ti
V = ∆d = d1 – d0
∆t t1 – t0
• Units are meters/second (m/s)
• Can be positive or negative depending on
direction moved.
v
d
The Signs of Velocity
First choose + direction; then
v is positive if motion is with
that direction, and negative if
it is against that direction.
Velocity is positive (+) or negative (-)
based on direction of motion.
- +
- +
+
example problem
During a race, Carla covers 650 m in 125 s
running east on a straight road. Find Carla’s
average velocity.
V = ∆d = 650 m = 5.2 m/s
∆t 125 s
How long will it take her to run 5 km?
5.2 m/s = .0052 km x 3600 s = 18.7 km
s hr hr
18.7 km = 5 km = .267 hr
hr x hr
Try one
• Heather and Matthew walk eastward with a speed of .98 m/s. If it takes them 34 min to walk to the store, how far have they walked?
• Knowns? What do you know? Write it down.
• Speed = .98 m/s, time = 34 minutes (2040 sec)
• Unknown? What do you want to know?
• How far? Distance = ?
• Equation? Write the equation you’ll use.
• Speed = distance / time
• Work the problem.
• .98 m/s = distance / 2040 sec; d = 2000 meters
Example. A runner runs 200 m, east, then changes
direction and runs 300 m, west. If the entire trip takes
60 s, what is the average speed and what is the
average velocity?
Recall that average speed is
a function only of total
distance and total time:
Total distance: s = 200 m + 300 m = 500 m
500 m
60 s
total pathAverage speed
time Avg. speed
8.33 m/s
Direction does not matter!
start
s1 = 200 m s2 = 300 m
Example 1 (Cont.) Now we find the average
velocity, which is the net displacement divided by
time. In this case, the direction matters.
xo = 0
t = 60 s
x1= +200 m xf = -100 m 0fx xv
t
x0 = 0 m; xf = -100 m
100 m 01.67 m/s
60 sv
Direction of final displacement
is to the left as shown.
Average velocity: 1.67 m/s, Westv
Note: Average velocity is directed to the west.
Example 2. A sky diver jumps and falls for 600 m
in 14 s. After chute opens, he falls another 400 m
in 150 s. What is average speed for entire fall?
625 m
356 m
14 s
142 s
A
B
600 m + 400 m
14 s + 150 s
A B
A B
x xv
t t
1000 m
164 sv 6.10 m/sv
Average speed is a function only
of total distance traveled and the
total time required.
Total distance/ total time:
Walking Trip Tracking speed and velocity
checking for understanding
An Indianapolis 500 car races around the track
at 225 mph. At the end of the race (500 miles),
what was its average velocity?
for example
A book gets pushed around the
perimeter of a table with
dimensions 1.75 m X 2.25 m. It
completes this motion in 23 s.
What is its average velocity?
What is its average speed?
another example
Car A travels from New York to
Miami at a speed of 25 m/s.
Car B travels from New York to
Canada at a speed of 25 m/s.
Are their velocities equal?
Problems
• You run down the road 500m. It takes you 32sec to complete the task.
• What is your:
– average speed?
– average velocity?
– displacement?
– distance?
• You run around a circular track (radius of 300m) in 32 sec
• What is your:
– average speed?
– average velocity?
– displacement?
– distance?
• What if you went half way?
• Will your average speed ever be zero?
one more for good measure
You travel on a straight highway from your house to visit
your friend 370 km (230 mi) to the west. You leave your
house at 10 am and arrive at 3 pm.
However, after you left your house, you realized you
forgot your toothbrush. You were only 15 km down the
road so you went back and got it.
Half way to your friend’s house, you took a short 5 km
side road to grab a burger at your favorite burger place.
What was your average velocity for this trip? What was your average speed for this trip?
Apply it
Could we determine the final position of
an object from its average velocity?
Show the equation you would derive to
solve this type of problem.
v = Δd vΔt = Δd Δd = df – di Δt so: df – di = vΔt df = vΔt
Calculating Velocity and Speed
acceleration The rate of changing velocity
Think about this....
What are three ways to change the velocity of a car?
Accelerate
Decelerate
Change direction
Slow your car to a stop at a stop sign. Slow from 9 m/s
to 0.0 m/s in 5 s.
Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s
in 1.5 s.
acceleration
The rate at which an object changes
its velocity in a given time.
• Has velocity so must include a direction.
• Any time an object’s velocity is changing,
the object has an acceleration.
Definition of Acceleration
An acceleration is the change in velocity per
unit of time. (A vector quantity.)
A change in velocity requires the application
of a push or pull (force).
A formal treatment of force and acceleration will be given later. For now, you should
know that:
• The direction of
acceleration is same as
direction of force.
• The acceleration is
proportional to the
magnitude of the force.
Change in velocity is often written as a = ∆v
∆t
If a car moves at a constant
velocity, then its
acceleration is zero
acceleration = change in velocity
change in time
acceleration (m/s2) = (vf) - (vi) time
©2008 by W.H. Freeman and Company
Acceleration
Acceleration is the rate of change of velocity.
t
v
tt
vva
12
12
©2008 by W.H. Freeman and Company
Acceleration
Acceleration is a vector, although in one-dimensional
motion we only need the sign.
The previous image shows positive acceleration; here is
negative acceleration:
calculating average acceleration
average acceleration = change in velocity
change in time
a = ∆ v units - v = (m/s) = m X 1 = m
∆ t t s s s s2
Constant Acceleration
Acceleration:
Setting to = 0 and solving for v, we have:
Final velocity = initial velocity + change in velocity
0fv v at
0
0
f
avg
f
v vva
t t t
try solving
Find the average velocity for the opening problem.
Slow your car to a stop at a stop sign. Slow from 9 m/s
to 0.0 m/s in 5 s.
Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s
in 1.5 s.
a = ∆ v - 9 m/s = - 1.8 m /s2 vf - vi
∆ t 5 s tf - ti
-9 m/s = - 6.0 m /s2
1.5 s
©2008 by W.H. Freeman and Company
Acceleration
There is a difference between negative acceleration and
deceleration:
Negative acceleration is acceleration in the negative direction as
defined by the coordinate system.
Deceleration occurs when the acceleration is opposite in direction to
the velocity.
- the sign of the velocity and the acceleration is the same if
the object is speeding up and that
- the sign of the velocity and the acceleration is the opposite
if the object is slowing down.
Review of Symbols and Units
• Displacement (x, xo); meters (m)
• Velocity (v, vo); meters per second (m/s)
• Acceleration (a); meters per s2 (m/s2)
• Time (t); seconds (s)
Review sign convention for each symbol
One last problem:
Will and grace enter a race. Both run at the same rate. Both
walk at the same rate. Will runs half the distance and walks
half the distance. Grace runs half the time and walks half
the time. Who wins?
Grace wins – she runs a longer distance. When she runs half
the time she covers more distance than when she walks half
the distance. So we know the she runs for more than half the
distance.
• Practice Problems
• pg. 59, 3.3 Section Review
• pg. 61, Chapter Review, problems 17-21
• Kinematics practice problems worksheet