Motion

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• Motion

2

• Describing Motion.

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• Change of position
• Passage of time
• Describe using a referent that is not moving.
• Change of position relative to some reference
  point (referent) over a defined period of time.

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The motion of this windsurfer, and of other
moving objects, can be described in terms of
the distance covered during a certain time
period.
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• Measuring Motion.

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• Speed.
• How much position has changed (displacement)
• What period of time is involved
• Speed distance/time
• Constant speed moving equal distances in equal
  time periods.
• Average speed average over all speeds including
  increases and decreases in speed.
• Instantaneous speed speed at any given instant.
• Considers a very short time period.
• v d/t

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This car is moving in a straight line over a
distance of 1 mi each minute. The speed of the
car, therefore, is 60 mi each 60 min, or 60
mi/hr.
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• Speed is defined as a ratio of the displacement,
  or the distance covered, in straight-line motion
  for the time elapsed during the change, or v
  d/t.
• This ratio is the same as that found by
  calculating the slope when time is placed on the
  x-axis.
• The answer is the same as shown on this graph
  because both the speed and the slope are ratios
  of distance per unit of time.
• Thus you can find a speed by calculating the
  slope from the straight line, or "picture," of
  how fast distance changes with time.

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If you know the value of any two of the three
variables of distance, time, and speed, you can
find the third. What is the average speed of this
car?
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• Example
• if one travels 230 miles in 4 hours, then the
  speed is
• v 230 mi. 57.5mi/hr
  4 hours

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• Velocity.
• Velocity IS A vector quantity, it describes both
  direction and speed.
• A quantity without direction is called a scalar
  quantity.

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Velocity is a vector that we can represent
graphically with arrows. Here are three
different velocities represented by three
different arrows. The length of each arrow is
proportional to the speed, and the arrowhead
shows the direction of travel.
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• Acceleration.
• Three ways to change motion.
• change the speed.
• change direction.
• change both speed and direction at the same time.
• a change of velocity over a period of time is
  defined as acceleration (rate).
• acceleration change of velocity
  time

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(A) This graph shows how the speed changes per
unit of time while driving at a constant 30 mi/hr
in a straight line. As you can see, the speed is
constant, and for straight-line motion, the
acceleration is 0.
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(B) This graph shows the speed increasing to 50
mi/hr when moving in a straight line for 5 s. The
acceleration, or change of velocity per unit of
time, can be calculated from either the equation
for acceleration or by calculating the slope of
the straight line graph. Both will tell you how
fast the motion is changing with time.
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Four different ways (A-D) to accelerate a car.
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• Example
• if a car changes velocity from 55 km/hr to 80
  km/hr in 5s, then
• acceleration80km/hr-55km/hr
  5s
• acceleration 5km/hr/s
• usually convert km/hr to m/s
• then get m/s2
• represented mathematically, this relationship is
• a Vf-Vi
  t

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• Aristotle's Theory of Motion.

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• Aristotle thought that there were two spheres in
  which the universe was contained
• The sphere of change
• The sphere of perfection

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• The Earth was seen to be a perfect sphere
• The Earth was fixed in place and everything else
  in the Universe revolved around the Earth

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One of Aristotle's proofs that the Earth is a
sphere. Aristotle argued that falling bodies
move toward the center of the Earth. Only if
the Earth is a sphere can that motion always be
straight downward, perpendicular to the Earth's
surface. Another of Aristotle's proofs that the
Earth is a sphere (A) The Earth's shadow on the
moon during lunar eclipses is always round. (B)
If the Earth were a flat cylinder, there would be
some eclipses in which the Earth would cast a
flat shadow on the moon.
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The shape of a sphere represented an early
Greek idea of perfection. Aristotle's grand
theory of the universe was made up of spheres,
which explained the "natural motion" of
objects.
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• Natural Motion
• Aristotle was the first to think quantitatively
  about the speeds involved in these movements. He
  made two quantitative assertions about how things
  fall (natural motion)
• Heavier things fall faster, the speed being
  proportional to the weight.
• The speed of fall of a given object depends
  inversely on the density of the medium it is
  falling through, so, for example, the same body
  will fall twice as fast through a medium of half
  the density.

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• Notice that these rules have a certain elegance,
  an appealing quantitative simplicity.
• And, if you drop a stone and a piece of paper,
  it's clear that the heavier thing does fall
  faster, and a stone falling through water is
  definitely slowed down by the water, so the rules
  at first appear plausible.
• The surprising thing is, in view of Aristotle's
  painstaking observations of so many things, he
  didn't check out these rules in any serious way.

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• It would not have taken long to find out if half
  a brick fell at half the speed of a whole brick,
  for example.
• Obviously, this was not something he considered
  important.
• From the second assertion above, he concluded
  that a vacuum cannot exist, because if it did,
  since it has zero density, all bodies would fall
  through it at infinite speed which is clearly
  nonsense.

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• Forced motion
• For forced motion, Aristotle stated that the
  speed of the moving object was in direct
  proportion to the applied force. (Aristotle
  called forced motion, violent motion).
• This means first that if you stop pushing, the
  object stops moving.

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• This certainly sounds like a reasonable rule for,
  say, pushing a box of books across a carpet, or a
  Grecian ox dragging a plough through a field.
• (This intuitively appealing picture, however,
  fails to take account of the large frictional
  force between the box and the carpet.
• If you put the box on a sled and pushed it across
  ice, it wouldn't stop when you stop pushing.
• Galileo realized the importance of friction in
  these situations.)

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• Forces.

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• A force is viewed as a push or a pull, something
  that changes the motion of an object.
• Forces can result from two kinds of interactions.
• Contact interactions.
• Interaction at a distance.

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• The net force is the sum of all forces acting on
  an object.
• When two forces act on an object the forces are
  cumulative (the are added together.
• Net force is called a resultant and can be
  calculated using geometry.

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• Four important aspects to forces.
• The tail of a force arrow is placed on the object
  that feels the force.
• The arrowhead points in the direction of the
  applied force.
• The length of the arrow is proportional to the
  magnitude of the applied force.
• The net force is the sum of all vector forces.

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The rate of movement and the direction of
movement of this ship are determined by a
combination of direction and magnitude of force
from each of the tugboats. A force is a vector,
since it has direction as well as magnitude.
Which direction are the two tugboats pushing?
What evidence would indicate that one tugboat is
pushing with greater magnitude of force? If the
tugboat by the numbers is pushing with a greater
force and the back tugboat is keeping the ship
from moving, what will happen?
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(A)When two parallel forces are acting on the
cart in the same direction, the net force is the
two forces added together.
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• (B) When two forces are opposite and of equal
  magnitude, the net force is zero.

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• (C) When two parallel forces are not of equal
  magnitude, the net force is the difference in the
  direction of the larger force.

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• You can find the result of adding two vector
  forces that are not parallel by drawing thetwo
  force vectors to scale, then moving one so the
  tip of one is the tail of the other.
• A new arrow drawn to close the triangle will tell
  you the sum of the two individual forces.

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(A) This shows the resultant of two equal 200 N
acting at an angle of 90O, which gives a
single resultant arrow proportional to a
force of 280 N acting at 45O. (B) Two unequal
forces acting at an angle of 60O give a single
resultant of about 140 N.
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• Horizontal Motion on Land.

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• It would appear as though Aristotle's theory of
  motion was correct as objects do tend to stop
  moving when the force is removed.
• Aristotle thought that the natural tendency of
  objects was to be at rest.
• Objects remained at rest until a force acted on
  it to make it move.

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• Aristotle and Galileo differed in how they viewed
  motion.
• Again, Aristotle thought that the natural
  tendency of objects was to be at rest.
• Galileo thought that it was every bit as natural
  for an object to be in motion.

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• Inertia.
• Galileo explained the behavior of matter to stay
  in motion by inertia.
• Inertia is the tendency of an object to remain in
  motion in the absence of an unbalanced force such
  as
• friction
• gravity.

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Galileo (left) challenged the Aristotelian view
of motion and focused attention on the concepts
of distance, time, velocity, and acceleration.
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• Falling Objects

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• Introduction
• The velocity of an object does not depend on its
  mass.
• Differences in the velocity of an object are
  related to air resistance.
• In the absence of air resistance (a vacuum) all
  objects fall at the same velocity.
• This is free fall which neglects air resistance
  and considers only the force of gravity on the
  object.

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• Galileo vs. Aristotle's Natural Motion.
• Galileo observed that the velocity of an object
  in free fall increased with the time the object
  was in free fall.
• From this he reasoned that the velocity would
  have to be
• proportional to the time of the fall
• proportional to the distance of the fall.

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According to a widespread story, Galileo dropped
two objects with different weights from the
Leaning Tower of Pisa. They were supposed to have
hit the ground at about the same time,
discrediting Aristotle's view that the speed
during the fall is proportional to weight.
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(A) This ball is rolling to your left with no
forces in thedirection of motion. The vector
sum of the force of floor friction (Ffloor)
and the force of air friction (Fair) result in
a net force opposing the motion, so the ball
slows to a stop.
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(B) A force is applied to the moving ball,
perhaps by a hand that moves along with the
ball. The force applied (Fapplied) equals the
vector sum of the forces opposing the motion, so
the ball continues to move with a constant
velocity.
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• Remembering the equation for velocity
• v d/ t
• We can rearrange this equation to incorporate
  acceleration, distance, and time.
• rearranging this equation to solve for distance
  gives.
• dvt
• An object in free fall should have uniform
  acceleration, so we can use the following
  equation to calculate the velocity.
• v vfvi
  2

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• Substitute this equation into dvt, we get
• d(vfvi) (t)
  (2)
• The intial velocity is zero so
• d(vf) (t)
  (2)

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• We can now substitute the acceleration equation
  in for velocity.
• a vf-vi
  t
• vfat
• d (at) (t)
  (2)
• Simplifying we get
• d1/2at2

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• Galileo knew from this reasoning that a free
  falling object should cover the distance
  proportional to the square of the time of the
  fall.
• He also knew that the velocity increased at a
  constant rate.
• He also knew that a free falling object
  accelerated toward the surface of the Earth.

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Galileo concluded that objects persist in their
state of motion in the absence of an unbalanced
force, a property of matter called inertia.
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Thus, an object moving through space without any
opposing friction (A) continues in a straight
line path at a constant speed. The application of
an unbalanced force in the direction of the
change, as shown by the large arrow, is needed to
(B) slow down, (C) speed up, or, (D) change the
direction of travel.
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• Acceleration Due to Gravity.
• Objects fall to the Earth with uniformly
  accelerated motion, caused by the force of
  gravity.
• All objects experience this constant
  acceleration.
• This acceleration is 9.8 m/s (32 ft/s) for each
  second of fall.
• This acceleration is 9.8 m / s2.
• This acceleration is the acceleration due to the
  force of gravity and is given the symbol g.

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An object dropped from a tall building covers
increasing distances with every successive second
of falling. The distance covered is proportional
to the square of the time falling (d ? t2).
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The velocity of a falling object increases at a
constant rate, 32 ft/s2.
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• Example
• A penny is dropped from the Eiffel Tower and hits
  the ground in 9.0 s. How far is if to the
  ground.
• d1/2gt2
• d1/2(9.8m/s2)(9.0s)2
• d(4.9m/s2)(27.0s2)
• d (m?s2)
  s2
• d m

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• Compound Motion.

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• Introduction.
• Three types of motion are
• Horizontal, straight line motion.
• Vertical motion due to the force of gravity.
• Projectile motion, when an object is thrown into
  the air by a given force.

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• Projectile motion can occur in several ways.
• Straight up vertically.
• Straight out horizontally.
• At some angle in between these two.
• Compound motion requires an understanding of the
  following
• Gravity acts on objects at all times, regardless
  of their position.
• Acceleration due to gravity is independent of the
  motion of the object.

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• Vertical Projectiles.
• When an object is thrown straight upward, gravity
  acts on it during its entire climb.
• Eventually, the force of gravity captures the
  object and it begins to fall to Earth with
  uniformly accelerated motion.
• At the peak of the ascent, it comes to rest and
  begins its acceleration toward the Earth with
  velocity of zero.

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On its way up, a vertical projectile such as a
misdirected golf ball is slowed by the force of
gravity until an instantaneous stop then it
accelerates back to the surface, just as another
golf ball does when dropped from the same height.
The straight up and down moving golf ball has
been moved to the side in the sketch so we can
see more clearly what is happening.
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• Horizontal Projectiles.
• Horizontal projectiles are usually projected at
  some angle and can be broken down into horizontal
  and vertical components.

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A horizontal projectile has the same horizontal
velocity throughout the fall as it accelerates
toward the surface, with the combined effect
resulting in a curved path. Neglecting air
resistance, an arrow shot horizontally will
strike the ground at the same time as one dropped
from the same height above the ground, as shown
here by the increasing vertical velocity arrows.
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A football is thrown at some angle to the
horizon when it is passed downfield.
Neglecting air resistance, the horizontal
velocity is a constant, and the vertical
velocity decreases, then increases, just as in
the case if a vertical projectile. The combined
motion produces a parabolic path. Contrary to
statements by sportscasters about the abilities
of certain professional quarterbacks, it is
impossible to throw a football with "flat
trajectory" because it begins to accelerate
toward the surface as soon as it leaves the
quarterback's hand.
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Without a doubt, this baseball player is aware of
the relationship between the projection angle and
the maximum distance acquired for a given
projection velocity.