– How much position has changed (displacement)
– What period of time is involved
– Speed = distance/time
– Constant speed = moving equal distances in equal
– 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
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
• 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.
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?
• if one travels 230 miles in 4 hours, then the
• v = 230 mi. = 57.5mi/hr
– Velocity IS A vector quantity, it describes both
direction and speed.
– A quantity without direction is called a scalar
Velocity is a vector that we can represent graphically
Here are three different velocities represented by three
The length of each arrow is proportional to the speed,
and the arrowhead shows the direction of travel.
– Three ways to change motion.
• change the speed.
• change direction.
• change both speed and direction at the same
– a change of velocity over a period of time is
defined as acceleration (rate).
– acceleration = change of velocity
(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.
(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.
Four different ways (A-D) to accelerate a car.
• if a car changes velocity from 55 km/hr to 80
km/hr in 5s, then
• acceleration = 5km/hr/s
• usually convert km/hr to m/s
• then get m/s2
• represented mathematically, this relationship is:
• a = Vf-Vi
• Aristotle thought that there were two
spheres in which the universe was
– The sphere of change
– The sphere of perfection
• 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
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
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.
The shape of
idea of perfection.
theory of the
made up of spheres,
the "natural motion"
• 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
– 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
• 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
– 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.
– It would not have taken long to find out if half a
brick fell at half the speed of a whole brick, for
– Obviously, this was not something he considered
– 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.
• 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,
– This means first that if you stop pushing, the
object stops moving.
– 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
• Galileo realized the importance of friction in
• 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.
• The net force is the sum of all forces acting on an
– 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.
• 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
– The length of the arrow is proportional to the
magnitude of the applied force.
– The net force is the sum of all vector forces.
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?
(A)When two parallel forces are acting on the cart in the
same direction, the net force is the two forces added
• (B) When two forces are opposite and of equal
magnitude, the net force is zero.
• (C) When two parallel forces are not of equal
magnitude, the net force is the difference in the
direction of the larger force.
• 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.
(A) This shows
the resultant of
two equal 200 N
acting at an
angle of 90O
which gives a
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.
• 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.
• Aristotle and Galileo differed in how they viewed
– 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.
– 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
Aristotelian view of
motion and focused
attention on the
concepts of distance,
time, velocity, and
– The velocity of an object does not depend on its
– 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.
• 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
– 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.
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
Aristotle's view that the
speed during the fall is
proportional to weight.
(A) This ball is
rolling to your
left with no
forces in the
vector sum of
the force of
(Ffloor) and the
force of air
result in a net force opposing the motion, so the ball
slows to a stop.
(B) A force is
applied to the
perhaps by a
hand that moves
along with the
ball. The force
equals the vector
sum of the forces opposing the motion, so the ball
continues to move with a constant velocity.
–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
• An object in free fall should have uniform
acceleration, so we can use the following
equation to calculate the velocity.
–v= vf+vi 2
•Substitute this equation into d=vt, we get
• The intial velocity is zero so:
•We can now substitute the acceleration
equation in for velocity.
–a= vf-vi t
–d= (at) (t)
• Simplifying we get
– 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
– He also knew that a free falling object accelerated
toward the surface of the Earth.
that objects persist
in their state of
motion in the
absence of an
unbalanced force, a
property of matter
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.
• Acceleration Due to Gravity.
– Objects fall to the Earth with uniformly
accelerated motion, caused by the force of
– 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.
dropped from a
second of falling.
the square of the
(d α t2
The velocity of
a falling object
increases at a
constant rate, 32
• A penny is dropped from the Eiffel Tower and
hits the ground in 9.0 s. How far is if to the
• d= (m•s2
• d= m
– 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.
–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
• Gravity acts on objects at all times, regardless
of their position.
• Acceleration due to gravity is independent of
the motion of the object.
• 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
– At the peak of the ascent, it comes to rest and
begins its acceleration toward the Earth with
velocity of zero.
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.
• Horizontal Projectiles.
– Horizontal projectiles are usually projected at
some angle and can be broken down into
horizontal and vertical components.
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
A football is thrown
at some angle to the
horizon when it is
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.
Without a doubt, this
baseball player is aware
of the relationship
between the projection
angle and the maximum
distance acquired for a