Post on 30-Mar-2018
transcript
Introduction to
Kinematics
Motion, Forces and Energy
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Mechanics: The study of motion
Kinematics The description of how things move 1-D and 2-D motion
Dynamics The study of the forces that cause motion Newton’s Laws Energy Momentum
Remember:
Before we can study something, scientists need to define
what it is they are studying. They determine what belongs
in the study boundaries – the system – all processes,
forces, and measurable properties.
Anything outside the system’s boundaries – the
surroundings.
The surroundings can affect the system – and these
interactions are studied as well.
1. Sketch the problem
Remember: problem solving method
3. Check it! 3. Calculate your Answer
2. Choose the formula(s)
Kinematics
What is Motion?
any physical movement or change in position or place, relative to a reference point
Referenceo
int
Movement
understanding vocabulary
A stationary landmark or
location from which we start or
make our observations.
Reference point
Choose the point of reference or reference frame that
is easiest for you to use in solving the problem or
answering the question.
Make all measurements from that point of reference or
reference frame.
understanding vocabulary
An unchanging rate of motion.
Moving at uniform speed.
understanding vocabulary
The rate of change of speed
during motion.
Choose the coordinates
Establish where “0” is (the origin)
Establish directions where the values increase
the origin (value of 0) is placed at the reference point.
understanding vocabulary
Imaginary line marked off in measured units
Using a coordinate system
example - football
What origins and direction are used to determine 1st down?
Length of punt?
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.
0 1 2 5 3 4
the particle model
What is the motion of the cart in this
diagram?
Slowing down (deceleration)
Change is position is less
per each time interval
time in seconds
We can use motion diagrams to
represent 4 concepts in kinematics:
Uniform motion (constant speed)
At rest
Speeding up (acceleration)
Slowing down (deceleration)
Demonstrate uniform and
accelerated motion
Design it, show it, use evidence to support it.
Write it up……..
1. Name, date, “Physical Science”, Title
2. Problem: How can I use a car & track to
demonstrate uniform motion AND accelerated
motion?
3. Diagram and label your set up
4. Collect data in a data table – multiple trials!
5. Graph your results
6. Conclusion: At least one paragraph.
Don’t just say your proved your
hypothesis – DESCRIBE HOW the
result answers the questions.
Collaborate to collect and analyze your data – but
everyone completes their own written lab report. Use
your own words – copying is not collaborating.
SCALARS AND VECTORS
Measuring motion in 1 and 2 dimensions
understanding vocabulary
A measurable quantity with
appropriate units. Can be positive,
negative or zero.
has magnitude
examples: temperature, volume, speed, mass, distance etc.
understanding vocabulary
A measurable quantity that also
includes direction. Can be positive or
negative.
has magnitude
AND
direction
examples:
try some….
a. 5 m See Answer
b. 30 m/sec, East See Answer
c. 5 mi., North See Answer
d. 20 degrees Celsius See Answer
e. 256 bytes See Answer
f. 4000 Calories
Scalar or Vector?
scalar
scalar scalar
scalar
vector
vector
Position vector
Proportional to the distance of the object from the
origin and points from the origin to the location of the
object at a particular time.
0
+
+ 1 2 4 5 3
meters
What is smiley’s position vector?
+ 3 m
Or - 3 m, right
origin
Describe the position of
the house relative to
the bicycle using only
scalars. 4 km
Describe the position of
the house relative to the
bicycle using vectors.
4 km, N
Is speed a scalar or
vector?
1. What is the bike’s speed
after 1 hour? 25 mi/hr
2. Does the bike speed up or
slow down at B? How do you
know? Speeds up, slope
increases
3. What is the speed at B?
B
C
50 mi/hr
4. What is the speed at C?
0, it is stopped
Distance, Direction and
Displacement How far – overall?
remember
“Motion occurs when an object’s position
changes during an interval of time.”
“One dimensional motion can be represented by
position along a single coordinate axis.”
understanding vocabulary
The length of a time span.
initial time - ti
final time - tf
tf - ti represents the time interval
use the formula: Δt = tf - ti
distance vs displacement
Distance is a scalar quantity. It is always positive.
It refers to how much ground an object has covered
during its motion. There is no direction associated with it.
Distance is the length of the actual path taken by an object. Consider travel from point A to point B in diagram below:
Distance d 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 d = 20 m
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.
distance vs displacement
Displacement is a vector quantity.
It is the object's overall change in position. It can be
positive or negative depending on your initial point of
reference.
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.
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 the diagram below, the origin, or initial position (do) is at 0.0 m. The
final position (d1) is at 50 m. The distance traveled from the origin is 50
m, but the displacement is 50 m to the right, or can be drawn similar
with a green line – vector.
We use arrows to show the direction from
the origin.
• 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
X 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 ______.
0 6 10 -2
remember:
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.
∆x = xf - xi
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
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
If displacement is the distance and direction
between 2 positions = change in position =
final position – initial position
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 physics, the movement from the origin is thought of as positive or
negative. In each case, an origin, starting place, or reference point
needs to be established. Then, it must be decided which directions
are considered positive and which is considered negative. Once a
group agrees on that, then you can determine displacement vectors.
For example, if we assume that up is positive, then Mt. Elinor, would
have an elevation displacement vector
of +2,400 ft, based off of the origin
of sea level. On the other hand,
Death Valley, California is below
sea level by 120 ft, so its
displacement would be -120 ft
compared to sea level.
Left:
Displacement is positive.
Right:
Displacement is negative.
displacement is written as:
When would position be – but
the displacement +?
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
Consider a car that travels 8 m, E then 12 m, W.
What is the car’s
displacement?
What is the distance
traveled? 20 m !!
12 m,W
D D = 4 m, W x
8 m,E
x = +8 x = -4
try a problem
• A person starts at the 5.0 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 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?
A
C
B 3 mi
4 mi Distance: AB + BC
3 + 4 = 7 mi
Displacement: measure of the hypotenuse
(AB)2 + (BC)2 = (AC)2
(3)2 + (4)2 = x2 9 + 16 = √25
= 5 mi., NE
Try some on your own
Graphing the Relationship (function)
Distance vs time graphs
We can represent motion
graphically in 1 D using a
motion dot diagram on the
X-axis.
We can represent distance
and displacement either in
1 D or in 2 D using
algebra and vectors.
• You drive 3 miles east and then 4 miles north.
What is your:
– Distance travelled?
– Displacement?
A
C
B 3 mi
4 mi
Distance: AB + BC
3 + 4 = 7 mi
Displacement: measure of the hypotenuse
(AB)2 + (BC)2 = (AC)2
(3)2 + (4)2 = x2 9 + 16 = √25
= 5 mi., NE
What if we want to show the
relationship between the motion and the
time it takes for that motion?
Distance vs time – graphing a linear function
When you create a ratio representing the
distance (or displacement) of an object and the
time it takes for that motion to occur, you are
creating a rate (a derived unit).
Ex: you travel to your house which is 5 miles away
in 30 min. at a constant rate of speed.
Ex: you travel to your house which is 5 miles away
in 30 min. at a constant rate of speed.
What is your rate of speed for this journey?
5 miles X 60 min. = 10 mi./hr
30 min 1 hr
What does this rate look like on a graph?
We can represent all kinds of motion graphically as
a comparison.
We will start with distance vs time graphs that
show the motion of the object during a certain
period of time.
The Moving Man
PHET simulations
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.
total distanceAverage speed
total time
dv
t
Speed
• The average speed of an object is defined as the total distance traveled divided by the total time elapsed
– Speed is a scalar quantity
Speed (meters/second) = distance (in meters) time (sec)
s = d t
Speed equation
Average speed totally ignores
any variations in the object’s
actual motion during the trip.
The total distance and the total
time are all that is important
calculate
s = = d
t
20 m
4 s
s = 5 m/s
Scalar – magnitude only, not direction dependent!
A
B s = 20 m
Time t = 4 s
1) Average speed – average of all your speeds over the whole trip. For
example, on a trip to Florida, (910 miles) the traveler took 13 hours to get there.
The average speed would be 70 mph.
This doesn’t mean that the car was going exactly 70 mi/hr the entire time.
Sometimes the car was going faster, and others slower.
2) Constant Speed - traveling at the same rate for a long period of time. Constant speed is having the cruise control on in the car. The car maintains the
same speed the entire time you are clocking it.
3) Instantaneous speed - rate at which an object is traveling at a certain
moment. This is your speedometer in your car. It tells you how fast the car is
going at the time you look at it.
There are three types of speed:
velocity
A quantity that measures how fast an
object’s position has moved 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
D=12 m
20o
Velocity
• It takes time for an object to undergo a displacement
• The average velocity is the rate at which the displacement occurs
• generally use a time interval, so ti = 0
f iaverage
f i
x xxv
t t tV = ∆d = d1 – d0
∆t t1 – t0
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)
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.
- +
- +
+
Velocity can be +/-
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.
Recap:
Speed vs. Velocity
• Cars on both paths have the same average velocity since they had the same displacement in the same time interval
• The car on the blue path will have a greater average speed since the distance it traveled is larger
example problem
During a race, Carla covers 650 m in 125 s
running east on a straight road. Find Carla’s
average speed.
s = 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 360 s = 1.87 km
s hr hr
1.87 km = 5 km = 2.67 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.
W E
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:
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?
?check your understanding
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?
explain
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?
acceleration
Think about this....
What are three ways to change the velocity of a car?
Speed up
Slow down
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 speed or velocity in a given time.
• If velocity - must include a direction (vector).
• Any time an object’s speed or velocity is
changing, the object has an acceleration.
If a car is moving at a
constant velocity, then
its acceleration is zero!
acceleration = change in velocity = ∆v
change in time ∆t
acceleration = (vf) - (vi) time elapsed t
v
tt
vva
12
12
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
Acceleration is represented in (m/s2)
In 1 D motion, we can use vector arrows (or a +/-
sign) to show that something is speeding up or
slowing down.
V is positive and A is negative
V is positive and A is positive
in one-dimensional motion we only need the sign
to show when something is speeding up or slowing
down.
here is slowing down:
- the sign of the velocity and the acceleration is the
same if the object is speeding up and
- the sign of the velocity and the acceleration is the
opposite if the object is slowing down.
Review of Symbols and Units
• Displacement meters (m)
• Velocity/speed; meters per second (m/s)
• Acceleration; meters per s2 (m/s2)
• Time; seconds (s)
Make sure you convert any units when solving
problems so they are all the same!
try solving
Find the acceleration from the first 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
another problem to solve
A meteoroid changed velocity from 1.0
km/s to 1.8 km/s in 0.03 seconds. What is
its acceleration?
26.7 m/s2 (1.8) – (1) =
0.03 s
While racing out of our school parking lot, I time myself at
a speed of 40 meters per second seven seconds after
starting. What was my acceleration during this time?
Acceleration = vf – vi
t2 – t1
(40 m/s) / 7 s = 5.7 m/s2
If I were to accelerate at this rate for another
ninety seconds, how fast would I be going?
a = v/t so v = at (5.7 m/s2)(97 sec) = 550 m/s
Try some on
your own…
What would a velocity vs time graph look like?
Graphing velocity and
acceleration
Moving Man #2
Maze Game