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Motion
Measuring Motion
• Speed
– Average Speed = distance covered / time taken
v = d/t
• metric unit of speed: m/s
• English unit of speed: ft/s
– Constant speed:
• moving equal distances in equal time periods
• an object covering 5 feet each second has a constant speed of
5ft/s
– If speed changes:
• Average speed: average over all speeds
• Instantaneous speed: speed at any given instant
Constant speed:
this car is moving in a straight line covering a
distance of 1 mi each minute. The car,
therefore, has a constant speed of 60 mi per
each 60 min, or 60 mi/hr.
Speed is the slope of the straight line graph of
distance (on the y-axis) versus time (on the xaxis)
Velocity
• shows how fast and in what direction an
object moves
• Velocity: speed + direction of motion
• it is a vector quantity
• vector: a quantity that has both magnitude
(size) and direction
– ex.: velocity, acceleration, force, etc.
• scalar: a quantity without direction (has
only magnitude)
– ex.: speed, time, distance, volume, surface area,
etc.
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.
Acceleration
– Three ways to change motion:
• change speed
• change direction
• change both speed and direction at the same time
– Average acceleration: change in velocity over the time taken to
make the change
vf – vi
vi = initial velocity
a=
t
vf = final velocity
t = time interval
– Metric unit for acceleration:
unit of velocity
unit of time
=
m/s
s
= m/s2
– English unit for acceleration: ft/s2
30 mi/hr
60
60 mi/hr
mi/hr
60 mi/hr
30 mi/hr
Four different ways (A-D) to accelerate a car.
(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 is the slope
of the straight line graph of speed (on the y-axis) versus time
(on the x-axis).
Forces
• result from two kinds of interactions:
– contact interactions
– interaction at a distance (ex.: gravitational force)
• force: a push or a pull
– changes the motion of an object
– it is a vector: has both magnitude and direction
Graphical Representation of a Force
• represented by an arrow
– 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
object
force
Adding Forces
• net force (resultant): the sum of all forces acting
on an object
• when two or more forces act on an object their
effects are cumulative (added together)
• forces are added considering:
– their directions
– their magnitudes (sizes)
• the net force (resultant) can be calculated using
geometry
10 lb west
(A)When two parallel forces are acting on the cart in the
same direction, the net force is the two forces added
together.
10 lb west
• (B) When two forces are opposite and of equal
magnitude, the net force is zero.
10 lb west
• (C) When two parallel forces are not of equal
magnitude, the net force is the difference and the
direction is that of the larger force.
Adding nonparallel forces
• To add two forces that are not parallel:
– draw the two force vectors to scale
– place the tip of one of the forces at the tail of the other
– draw a vector to close the triangle: this is the net force
(the vector sum of the two forces)
vector sum
F1
F2
(A) This shows two equal forces (200 N each) acting at an angle
of 90O, which give a resultant force (Fnet) of 280 N acting at 45O.
(B) Two unequal forces acting at an angle of 60O give a single
resultant of about 140 N.
Inertia
• Galileo:
– natural tendency of objects:
• at rest
• in motion: explained the behavior of matter to
stay in motion by inertia
– inertia: natural tendency of an object to
remain at rest or in unchanged motion in the
absence of any forces
(A) This ball is
rolling to your
left with no forces
in the direction 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.
(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.
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 as
shown by the large
arrow, is needed to (B)
slow down, (C) speed
up, or, (D) change the
direction of travel.
Galileo (1564-1642)
challenged the
Aristotelian view of
motion and focused
attention on the
concepts of distance,
time, velocity, and
acceleration
Falling Objects
• free fall : due to force of gravity on the object
• the velocity of a falling object does not depend on
its mass
• in the absence of air resistance (in a vacuum) all
objects fall at the same velocity
– differences in the velocities of falling objects are due to
air resistance
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.
The actual
leaning tower of
Pisa taken by
my friend Larry
Heath, Emeritus
Professor of
Technology on
his recent trip to
Rome
Acceleration Due to Gravity:
g = 9.8 m/s2 = 32 ft/s2
• Free Fall:
– at a constant acceleration caused by the force
of gravity
– all objects experience this constant acceleration
– this acceleration is 9.8 m/s2 or 32 ft/s2
– This means that the velocity of a free falling
object increases at a constant rate (i.e., by 9.8
m/s every one second, or by 32 ft/s every one
second)
The velocity of
a falling object
increases at a
constant rate
(i.e., by 32 ft/s
each second)
– remember the equation for velocity:
v = d/ t
– can be rearranged to incorporate acceleration, distance,
and time.
• solve for distance:
d = vt
– an object in free fall has uniform (constant) acceleration,
so we can calculate the average velocity as:
vi + vf
v= 2
– substitute this equation into d = vt to get
(vi + vf) (t)
d=
2
• vi = 0 (for a free falling object)
(vf) (t)
d=
2
• use the acceleration equation:
a=
vf - vi
t
but vi = 0 , which gives a = vf /t
Solve for vf :
vf = at
• Substituting vf in the equation for d above, we get:
(at) (t)
d=
2
d = (1/2)at2 which gives the distance covered in the
free fall
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).
Projectile motion
• when an object is thrown into the air by a
given force
• projectile motion can be:
• straight up vertically
• object thrown straight out horizontally
• object thrown at some angle in between these two
• in a projectile motion:
• gravity acts on objects at all times (regardless of
their position)
• acceleration due to gravity is constant and
independent of the motion of the object
Vertical Projectiles
• an object thrown straight up into the air
• gravity acts on the object at all times,
pulling it down
– as the object moves up its velocity decreases
(gravitational force slows down the object)
– at the peak of the ascent, the object comes to
rest (for an instant) and begins its fall toward
the Earth
– during the fall its velocity increases at a
constant rate (i.e., acceleration is constant and
equal to g, acceleration due to gravity).
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
• an object thrown straight out horizontally
– the force of gravity acts on the object at all
times, pulling it down
• the motion of the object can be broken
down into two components:
– a horizontal motion at constant velocity
– a vertical motion with constant acceleration
(i.e., increasing velocity)
A horizontal projectile has a constant horizontal velocity and an
increasing vertical velocity as it falls to the ground. The combined effect
of the two velocities results in a curved path (parabola). 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.
Projectile
thrown
at an angle
A football is thrown at some angle. Neglecting air resistance, the
horizontal velocity is constant, and the vertical velocity decreases (on
the way up) then increases (on the way down), just as in the case of a
vertical projectile. The combined motion produces a parabolic path.