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Chapter Two Lecture
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What is Motion?
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 Formulas:
Constants:
g = -9.80 m/s2
Demonstrations:
- Drop balls of different weights.
- Hinged board to show center of mass and constant acceleration.
Main Ideas:
- Define various physical quantities relating to motion.
- Understand uniform accelerated motion (including falling bodies).
- Learn how to solve problems.
- Understand graphical analysis of motion.
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Definitions
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Frame of Reference
Any measurement of position, distance, and/or speed must be made from an "origin
," which is a starting point from which to take further observations and measure, collect, and analyze our experiences. Imagine two people standing at opposite ends of a room with a cat exactly
between them. If the cat walked toward one of the people, do you suppose they would describe the cat's motion in exactly the same way?
In fact, one person would say, "The cat walked toward me," but the other person would say, "The cat walked away from me." Two people watching
the same event would describe it differently because they witness it from different points of view. We need to be very careful that when we describe the motion of things, we do so in a way that is unambiguous;
we need to make clear our perspective.
Distance and Displacement
Distance is the total length that an object has moved. If I walk 2 km
north, then 1 km south, the distance I have moved is 3 km. On the other hand displacement is the net change in position. My displacement is
only 1 km. 3 west, 4 north: distance = 7, displacement = 5. For displacement;
Dx = xf - xo. ( Joe! - explain and draw x axis).
Speed
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Problem: If a dog jogs for 1.5 hours at an average speed of 2.22m/s, how
far does he go? How far means what is the distance he goes. We do algebra to get d = st.
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Velocity
Average velocity is the displacement/(elapsed time). The rate over which displacement changes:
If it takes me 45 minutes to walk the two km west, then one km east, my average speed is 3.0/0.75 = 4.0 km/hour while my average velocity is
1.0/.75 = 1.3 km/hour.
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 Problem: The land speed record was set in a jet powered
car in 1983 and was 283 m/s (633 miles/hour). The record is computed by driving 604 m in
one direction, then turning around and doing the same. If his first pass is done in 2.12 s and the second in 2.15 s, what is his average
velocity on each pass?
v =Dx/Dt = +604/2.12 = +285 m/s
v =Dx/Dt = -604/2.15 = -281 m/s (Average is 283 m/s).
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Instantaneous Velocity
Instantaneous velocity is the velocity at an instant. It is defined as
the average velocity over a very short period of time.
Your car speedometer gives your instantaneous speed, but not instantaneous velocity. Why?
Vectors and Scalars – Where would you like to go today?
In our study of physics we come across two kinds of information. The
first kind is something that can be described by a single number like the area of a circle or the speed or the distance. This is called a scalar. We
will also come across quantities that have two pieces of information, a magnitude and a direction. Velocity is such a quantity. It has a
magnitude, say 50 miles/hour, and a direction, say northwest. We write scalars simply as numbers because that is what they are. We write
vectors as bold face or with an arrow over them. Every vector quantity must have a magnitude and a direction. If I ask for a vector quantity on a
problem, both must be given or the problem is only half correct.
Acceleration
When the velocity of an object changes (any kind of change), we say the object accelerates
. There are many ways for acceleration to occur:
- If a moving object's speed increases, even if its direction stays constant, the object has accelerated.
- If a moving object's speed decreases, even if its direction stays constant, the object has accelerated.
- If a moving object's direction changes, even if its speed stays constant, the object has accelerated.
- If a stationary object begins moving, the object has accelerated.
When a person is driving straight down the road at 20 mph and
presses down more on the gas pedal increasing the velocity of the car to 50 mph, the car experiences an acceleration. Surely you agree with this.
The following are also examples of acceleration, although they may not be commonly described as such. When a person is driving straight down
the road at 50 mph and presses firmly on the brake pedal to reduce the car's velocity to 20 mph, the car experiences an acceleration because
there was a change in velocity. When a person is driving at a constant speed of 30 mph and only turns the steering wheel to alter the direction the car is moving, the car still experiences an acceleration. Why?
Because a change in direction of a moving object is a change in the object's velocity! Any kind of change in an object's velocity causes the object to experience an acceleration.
Average Acceleration
Average acceleration is given by
Note that an object is accelerating if it changes speed or if changes
direction since acceleration and velocity are vectors. Acceleration is the rate at which velocity changes, while velocity is the rate at which position changes.
Instantaneous Acceleration
Instantaneous acceleration is given by
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Problem: A sports car accelerates to the East from 0 to 60 mph in 6.0
seconds. What is its average acceleration in units of m/s2?
a = 10 miles per hour/second = 10 mph/s in an Eastward direction
This means that for every second of time the car's velocity changes by
10 mph. But, this is not how the answer is usually written, nor what the question asks. The units of measure must be consistent. So
remembering that 1 hour = 3600 s and 1 mile = 1600 m we have;
 = 4.5 m/s2 to the East. |
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Problem: I am driving 20 m/s to the East when a dog runs across my
path. I slam on my brakes and in 2 seconds slow to 5 m/s to the East. What was my average acceleration?
Using the definition of average acceleration;
we can plug in given values.
The negative sign means that the acceleration is to the West, so, by
leaving the velocities given as positive, we assume that East is the positive direction, and West is the negative direction.
As the car is moving with a velocity to the East, the acceleration is in
the opposite direction, when the two are opposite in direction the object is "slowing down."
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Uniform Acceleration
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We now use these simple definitions to derive equations for the
special case when acceleration is a constant (Elaborate on what that means). We start with the definition of acceleration and set to = 0 seconds and tf = t.
 What does this mean?
Setting to = 0, we get average velocity
and with
The last set of equations is for the special case of a motion with a constant acceleration only.
Now we can play around with more of these equations. Since
, we substitute this in one of the previous equations to get
 plugging in for vf we get
or
Finally, from substituting and expressions for "t" into the same equation we can get something even more interesting.
with an expression for time,
 plugged in
 and a little bit of multiplication of terms becomes
 and with a bit of algebraic rearranging becomes
We have derived the kinematic equations of motion with constant acceleration (i.e. any motion that takes place with a constant change in velocity).
Kinematic Equations of Motion
This table of the equations illustrates the variables in use for each.
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xf , xo
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vo
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vf
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a
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t
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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*
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These only work when acceleration is a constant (Magnitude and
Direction). If you are given three known quantities and one unknown, you can chose one of these equations. If you are given two known quantities and two unknown ones, you will have to use two of these
equations.
"Solving Problems"
- Read the Problem Carefully.
- Draw a Diagram
- Write down what is known or given and what you want to know.
- Think about the physics principles and make sure the equations are valid.
- Do the calculation.
- Think about the answer. Is it reasonable? (Order of magnitude)
- Check the units.
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Problem: Previously, we had seen that our sports car could accelerate from
rest at 4.5 m/s2. (This is a constant acceleration). After 8 seconds how far has it gone? We have a, t, and vo, and we want xf – xo so we can use
 to the East (according to the picture) |
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Problem: What is the speed of the car after 10 seconds?
We have a, t, and vo, and we want v, so we use
 (or about 100 miles/hour). |
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NOTE:
We often use the word deceleration to mean that an object is slowing
down. What is deceleration? Is it the same as negative acceleration? NO! Negative acceleration means the sign of the acceleration is negative.
Deceleration is when the direction of acceleration is opposite the direction of motion. Suppose a car is traveling in the negative direction.
To decelerate it must have an acceleration in the positive direction. On the other hand it will gain speed when the acceleration is in the negative direction.
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Example: A car is traveling in the negative direction at -32 m/s. The driver
applies his brakes and stops in 7.1 seconds. What was the car's acceleration? We have vo, vf, and t and we want a.
The change in velocity is (vf - vo); 0 m/s – (-32 m/s) = 32 m/s; so,
so even though "a" is positive, the car is decelerating because the
acceleration is in the opposite direction from the motion.
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Free Falling Bodies
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 One of the most important cases of uniform acceleration involves objects near earth that are allowed to fall. At first, it might seem that different
objects would accelerate at different rates depending on their weights. In fact, for thousands of years, people believed that heavier objects would fall faster
than lighter objects. For example, if a person had a heavy rock and a grape and dropped them from the same height at the same moment, people believed the heavy rock would hit the ground much sooner than the
grape. Say the grape was 1/100th the weight of the rock and both were dropped from a height of 1 meter. People believed the grape would fall 1 cm in the time the rock fell 1
meter, that is, the rock would fall 100 times more quickly. However, in the 1500s, the Italian scientist, Galileo Galilei showed that objects of
different weights dropped at the same moment, fall at the same rate. You can demonstrate this for yourself. Take a heavy and light object that are about the same shape and drop them from the same height at the
same time. You will notice that the heavy object and the light object fall through the same height in the same amount of time.
You might already be thinking about exceptions to this idea that
objects of different weights fall at the same rate. Consider a piece of paper and a rock. If you drop a heavy rock and a flat piece of paper at the same time, the rock would fall noticeably faster than the paper,
which would slowly flutter to the ground. This observation seems to offer evidence that contradicts Galileo's principle. However, take that same
piece of paper, crumple it into a ball and try dropping it with the rock again. Both now fall at about the same rate. Why does the crumpled
paper fall differently from the flat piece that flutters to the ground? It is because air resistance pushes against the moving object. Air resistance
is the resistance to the motion of an object moving through the air due to the object's collisions with numerous molecules of air. If there were no
air resistance then even a piece of open paper would drop at the same rate as a rock. Galileo proposed that, in the absence of air, objects of any size and shape fall at the same rate.
In the absence of air – a vacuum – when an object is dropped on the
Earth, its velocity increases by 9.8 m/s every second - that is, it accelerates 9.8 m/s2. If there were no air resistance, this acceleration,
which is called the acceleration due to gravity, would continue every second. On the moon, where there is no air, a feather and a hammer fall at the same rate – about 1.6 m/s every second.
For many applications we can neglect air resistance - then everything
falling near the surface of the earth accelerates toward the center of the earth at 9.8 m/s2. Therefore, all of the equations for constant (uniform)
acceleration apply to an object in free fall, (i.e. fall that neglects air resistance). This constant acceleration of about 9.80 m/s2 is called the
acceleration due to gravity, g. Also you should realize that the acceleration due to gravity is a vector quantity and we need to give it direction like "down." Often down is called the "negative" direction, then
g = -9.8 m/s2. The acceleration due to gravity is constant and will not change the entire time an object is freely falling.
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Example: A stone is dropped from rest from a very tall building. What is its
position and velocity after 3.0 seconds?
 a = g = -9.8 m/s2
vo = 0 m/s (rest)
t = 3.0 seconds
What does the minus mean in front of 44.1m? The stone fell "down"
44.1 meters or it lost 44.1 m of height; so – 44.1m
Again, the minus means the stone is falling downward. |
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Example: Suppose a spacecraft is traveling
with a speed of 3250 m/s, and it slows down by firing its retro rockets, so that a=-10 m/s2. What is the velocity of the spacecraft
after it has traveled 215 km? We know vo, a, and xf – xo, and we want vf so we use the last equation.
Problem: This is a harder one. Suppose a slow Dallas
Cowboy lineman picks up a fumbled football on the 20-yard line and runs toward the end zone at 7.3 m/s. The safety is
standing on the 23 yard line and needs to catch up to the lineman before he scores a touchdown. If the safety can
accelerate at a constant rate, what must be his minimum acceleration to catch the lineman? What will be the safety's final velocity?
Example: A boy throws a ball
upward from the top of a building with an initial velocity of 20.0 m/s. The building is 50 meters high. Determine (a) the time eeded
for the stone to reach its maximum height (b) the maximum height (c) the time needed for the stone to reach the level of the thrower (d)
the velocity of the stone at this instant 
