Transcript Graphing Ideas in Physics

Graphing Ideas in Physics
And Use of Vectors
1.4 Simple Types of Motion
• Below is a table of values of time and distance.
• The table shows the values of distance (d) at
certain times (t).
– These values all satisfy the equation d = 7t.
– You could make the table longer or shorter by
using different time increments.
1.4 Simple Types of Motion
• A way to show the
relationship between
distance and time is to
graph the values in the
table.
• The usual practice is to
graph distance versus
time, which puts distance
on the vertical axis.
– Note that the data points
lie on a straight line.
1.4 Simple Types of Motion
• Remember this simple rule: when the speed is
constant, the graph of distance versus time is
a straight line.
– In general, when one quantity is proportional to
another, the graph of the two quantities is a
straight line.
1.4 Simple Types of Motion
• An important feature of this graph is its slope.
– The slope of a graph is a measure of its steepness.
– In particular, the slope is equal to the rise between
two points on the line divided by the run between
the points.
1.4 Simple Types of Motion
• This is illustrated in the figure shown.
– The rise is a distance, Dd, and the run is a time
interval, Dt.
– So the slope equals Dd
divided by Dt, which is
also the object’s
velocity.
• The slope of a distanceversus-time graph
equals the velocity.
1.4 Simple Types of Motion
• The graph for a faster-moving body, a
racehorse for instance, would be steeper—it
would have a larger slope.
• The graph of d versus t for a slower object (a
person walking) would have a smaller slope.
• When an object is standing still (when it has
no motion), the graph of d versus t is a flat line
parallel with the horizontal axis.
– The slope is zero because the velocity is zero.
1.4 Simple Types of Motion
• Even when the velocity is not constant, the
slope of a d versus t graph is still equal to the
velocity.
– In this case, the graph is not a straight line
because, as the slope changes (a result of the
changing velocity), the graph curves or bends.
1.4 Simple Types of Motion
• The graph shown represents the motion of a
car that starts from a stop sign, drives down a
street, and then stops and backs into a parking
place.
1.4 Simple Types of Motion
• When the car is stopped, the graph is flat.
– The distance is not changing and the velocity is zero.
• When the car is backing up, the graph is
slanted downward.
– The distance is decreasing, and the velocity is
negative.
1.4 Simple Types of Motion
Constant Acceleration
• Keep in mind that the graph of distance versus
time is a straight line for uniform motion.
– For constant acceleration it is the graph of velocity
versus time that exhibits a linear behavior.
1.4 Simple Types of Motion
Constant Acceleration
• As shown in the table of distance values for a
falling body, distance increases rapidly.
1.4 Simple Types of Motion
Constant Acceleration
• The graph of distance versus time curves
upward.
– This is because the velocity of the body is
increasing with time, and the slope of this graph
equals the velocity.
1.4 Simple Types of Motion
• Rarely does the acceleration of an object stay
constant for long.
– As a falling body picks up speed, air resistance
causes its acceleration to decrease.
– When a car is accelerated from a stop, its
acceleration usually decreases, particularly when
the transmission is shifted into a higher gear.
1.4 Simple Types of Motion
• The figure shows the
velocity of a car as it
accelerates from 0 to
80 mph.
– Note that acceleration
steadily decreases (the
slope gets smaller).
– During the short time
the transmission is
shifted, the
acceleration is zero.
1.4 Simple Types of Motion
During a karate demonstration,
a concrete block is broken
by a person’s fist.
1.4 Simple Types of Motion
• The fist travels downward
until it contacts the block,
at about 6 milliseconds.
– This causes a large
acceleration as the fist is
brought to a sudden stop.
1.4 Simple Types of Motion
• The graph shows the
velocity of the fist, just
the slope of the
distance versus time
graph at each time.
– Contact with the
concrete is indicated by
the steep part of the
graph as the velocity
goes to zero.
1.4 Simple Types of Motion
• If we take the slope of
this segment of the
velocity graph, we find
that the magnitude of
the acceleration of the
fist at that moment is
about 3,500 m/s2, or
360g (ouch!).
– What happened at about
25 milliseconds?
2-8 Graphical Analysis of Linear
Motion
This is a graph of x vs. t for an object moving
with constant velocity. The velocity is the slope
of the
x-t curve.
© 2014 Pearson Education, Inc.
2-3 Instantaneous Velocity
On a graph of a particle’s position vs. time, the
instantaneous velocity is the tangent to the curve at
any point.
Concept Map 1.2
3-1 Vectors and Scalars
A vector has magnitude as well as
direction.
Some vector quantities:
displacement, velocity, force,
momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time,
temperature
Vectors
•Quantities that have both a magnitude and
a direction.
–Magnitude consists of an amount and the units.
–Direction can be expressed as some defined "x-hat"
or "y-hat" direction or as East, North, etc.
–Example:
–A displacement vector (distance with direction):
40 meters East
Drawing Vectors
We draw a vector as:
An arrow from tail to head with the head of the
arrow pointing in the direction of the vector.
The length of the arrow represents the vector's
magnitude
We specify its direction by the angle it makes with
the (+) horizontal axis.
tail
head
3-2 Addition of Vectors—Graphical
Methods
For vectors in one dimension,
simple addition and
subtraction are all that is
needed.
You do need to be careful
about the signs, as the figure
indicates.
Adding Vectors
• To Add a Set of Vectors
Draw the first vector from a start point on a grid
with horizontal and vertical axes.
Draw the next vector starting from the head of
the previous vector.
Repeat this last step until the last vector is
drawn.
Now draw the resultant (or “net”)vector starting
from the tail of the first vector to the head of the
last vector (or in other words: from start point to
end point).
1.2 Speed and Velocity
Vector Addition
• Sometimes a moving body has two velocities
at the same time.
– The runner on the deck of the ship in the figure
has a velocity relative to the ship and a velocity
because the ship itself is moving.
1.2 Speed and Velocity
Vector Addition
– A bird flying on a windy day has a velocity relative
to the air and a velocity because the air carrying
the bird is moving relative to the ground.
• The velocity of the runner relative to the
water or that of the bird relative to the ground
is found by adding the two velocities together
to give the net, or resultant, velocity.
• Let’s consider how two velocities (or two
vectors of any kind) are combined in vector
addition.
1.2 Speed and Velocity
Vector Addition
• When adding two velocities, you represent
each as an arrow with its length proportional
to the magnitude of the velocity—the speed.
– For the runner on the ship, the arrow representing
the ship’s velocity is a little more than 2 times as
long as the arrow representing the runner’s
velocity because the two speeds are 20 mph and 8
mph, respectively.
1.2 Speed and Velocity
Vector Addition
• Each arrow can be moved around for
convenience, provided its length and its
direction are not altered.
– Any such change would make it a different vector.
1.2 Speed and Velocity
Vector Addition
• The procedure for adding two vectors is as
follows.
– Two vectors are added by representing them as
arrows and then positioning one arrow so its tip is
at the tail of the other.
– A new arrow drawn from the tail of the first arrow
to the tip of the second is the arrow representing
the resultant vector—the sum of the two vectors.
1.2 Speed and Velocity
Vector Addition
• The figure shows this for the runner on the
deck of the ship.
• The runner is running forward in the direction
of the ship’s motion, so the two arrows are
parallel.
– When the arrows are positioned “tip to tail,” the
resultant velocity vector is parallel
to the others, and its
magnitude—the speed—is
28 mph (8 mph +20 mph).
1.2 Speed and Velocity
Vector Addition
• In this figure, the runner is running toward the
rear of the ship, so the arrows are in opposite
directions.
– The resultant velocity is again
parallel to the ship’s velocity,
but its magnitude is 12 mph
(20 mph – 8 mph).
1.2 Speed and Velocity
Vector Addition
• Vector addition is done the same way when the
two vectors are not along the same line.
• The figure shows a bird with velocity 8 m/s north
in the air while the air itself has velocity 6 m/s
east.
1.2 Speed and Velocity
Vector Addition
• The bird’s velocity observed by someone on
the ground, (b), is the sum of these two
velocities.
1.2 Speed and Velocity
Vector Addition
• We determine this by placing the two arrows
representing the velocities tip to tail as before
and drawing an arrow from the tail of the first
to the tip of the second.
– The direction of the resultant velocity is toward
the northeast.
3-2 Addition of Vectors—Graphical Methods
If the motion is in two dimensions, the situation is somewhat more
complicated.
Here, the actual travel paths are at right angles to one another; we
can find the displacement by using the Pythagorean Theorem.
3-2 Addition of Vectors—Graphical Methods
Adding the vectors in the opposite order gives the
same result:
1.2 Speed and Velocity
Vector Addition
• Watch for this when you see a bird flying on a
windy day:
– Often the direction the bird is moving is not the
same as the direction its body is pointed.
1.2 Speed and Velocity
Vector Addition
• What about the magnitude of the resultant
velocity?
• It is not simply 8 + 6 or 8 – 6, because the two
velocities are not parallel.
– With the numbers chosen for this example, the
magnitude of the resultant velocity—the bird’s
speed—is 10 m/s.
1.2 Speed and Velocity
Vector Addition
• If you draw the two original arrows with correct
relative lengths and then measure the length of
the resultant arrow, it will be 5/4 times the
length of the arrow representing the 8 m/s
vector.
– Then 8 m/s times 5/4 equals 10 m/s.
– This can be calculated using the Pythagorean
theorem because the arrows form a right triangle.
1.2 Speed and Velocity
Vector Addition
• Vector addition is performed in the same
manner, no matter what the directions of the
vectors.
• The figure shows two other examples of a bird
flying with different wind directions.
– The magnitudes of the resultants in these cases
are best determined by measuring the lengths of
the arrows.
1.2 Speed and Velocity
Vector Addition
• There are many other situations in which a
body’s net velocity is the sum of two (or more)
velocities (for example, a swimmer or boat
crossing a river).
• Displacement vectors are added in the same
fashion.
– If you walk 10 meters south, then 10 meters west,
your net displacement is 14.1 meters southwest.
1.2 Speed and Velocity
Vector Addition
• The process of vector addition can be “turned
around.”
• Any vector can be thought of as the sum of
two other vectors, called components of the
vector.
– When we observe the bird’s single velocity, we
would likely realize that the bird has two velocities
that have been added.
1.2 Speed and Velocity
Vector Addition
• A soccer player running southeast across a
field can be thought of as going south with
one velocity and east with another velocity at
the same time.
3-2 Addition of Vectors—Graphical
Methods
Even if the vectors are not at right angles, they
can be added graphically by using the tail-to-tip
method.
3-2 Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here again
the vectors must be tail-to-tip.
3-3 Subtraction of Vectors, and Multiplication
of a Vector by a Scalar
In order to subtract vectors, we define the negative
of a vector, which has the same magnitude but
points in the opposite direction.
Then we add the negative vector.
3-3 Subtraction of Vectors, and Multiplication
of a Vector by a Scalar
A vector can be multiplied by a scalar c; the result is a
vector c that has the same direction but a magnitude
cV. If c is negative, the resultant vector points in the
opposite direction.
V
V
3-4 Adding Vectors by Components
Any vector can be expressed as the sum of two other
vectors, which are called its components. Usually the other
vectors are chosen so that they are perpendicular to each
other.
3-4 Adding Vectors by Components
If the components are
perpendicular, they can be
found using trigonometric
functions.
3-4 Adding Vectors by Components
The components are effectively one-dimensional, so they can be
added arithmetically.
3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
and
.
3-4 Adding Vectors by
Components
Example 3-2: Mail carrier’s
displacement.
A rural mail carrier leaves the post
office and drives 22.0 km in a
northerly direction. She then drives
in a direction 60.0° south of east for
47.0 km. What is her displacement
from the post office?
3-4 Adding Vectors by
Components
Example 3-3: Three short trips.
An airplane trip involves three legs, with
two stopovers. The first leg is due east for
620 km; the second leg is southeast (45°)
for 440 km; and the third leg is at 53°
south of west, for 550 km, as shown. What
is the plane’s total displacement?
3-5 Unit Vectors
Unit vectors have magnitude 1.
V
Using unit vectors, any vector
can be written in terms of its components:
3-6 Vector Kinematics
In two or three dimensions,
the displacement is a vector:
3-6 Vector Kinematics
As Δt and Δr become smaller and
smaller, the average velocity
approaches the instantaneous
velocity.
3-6 Vector Kinematics
The instantaneous acceleration is
in the direction of Δ = 2 – 1, and
is given by:
v
v
v
2-9 Graphical Analysis and Numerical
Integration
The total displacement of an object
can be described as the area under
the v-t curve: