1/30/14 Agenda
Chapter 2, Problem 14
•  Today: Homework #2 Quiz, Finish free fall,
more vectors
•  Finish reading Chapter 3 by Thursday
•  Thursday: 2D motion & projectiles
Free Fall – an object dropped
•  Initial velocity is zero
•  Let up be positive
•  Use the kinematic
equations
–  Generally use y
instead of x since
vertical
vo= 0
a = -g
In an 8.00 km race, one runner runs at a
steady 11.0 km/h and another runs at 14.0
km/h. How far from the finish line is the
slower runner when the faster runner
finishes the race? Show all your work. You
may leave your answer in km.
Free Fall – object thrown
downward
•  a = -g = -9.80 m/s2
•  Initial velocity ≠ 0
– With upward being
positive, initial velocity
will be negative
•  Acceleration is -g =
-9.80 m/s2
Free Fall – object thrown upward
•  Initial velocity is
upward, so positive
•  The instantaneous
velocity at the
maximum height is
zero
•  a = -g = -9.80 m/s2
everywhere in the
motion
v=0
Thrown upward, cont.
•  The motion may be symmetrical
– Then tup = tdown
– Then vf = -vi
•  The motion may not be symmetrical
– Break the motion into various parts
•  Generally up and down
1 1/30/14 Non-symmetrical
Free Fall
•  Need to divide the
motion into
segments
•  Possibilities include
–  Upward and
downward portions
–  The symmetrical
portion back to the
release point and
then the nonsymmetrical portion
Properties of Vectors
•  Equality of Two Vectors
–  Two vectors are equal if they have the same
magnitude and the same direction
•  “Movement” of vectors in a diagram
–  Any vector can be moved parallel to itself
without being affected
Adding Vectors
•  When adding vectors, their directions must
be taken into account
•  Units must be the same
•  Geometric Methods
–  Use scale drawings
•  Algebraic Methods
–  More convenient
Vector vs. Scalar Review
•  All physical quantities encountered in this
text will be either a scalar or a vector
•  A vector quantity has both magnitude
(size) and direction -> vel,, accel., disp.
•  A scalar is completely specified by only a
magnitude (size) -> time, speed, dist.
More Properties of Vectors
•  Negative Vectors
–  Two vectors are negative if they have the
same magnitude but are 180° apart (opposite
directions)
–  A = -B; A + (-A) = 0
•  Resultant Vector
–  The resultant vector is the sum of a given set
of vectors
–  R = A + B
Adding Vectors Geometrically
(Tip-to-tail method)
•  Choose a scale
•  Draw the first vector with the appropriate length
and in the direction specified, with respect to a
coordinate system
•  Draw the next vector with the appropriate length
and in the direction specified, with respect to a
coordinate system whose origin is the end of
vector A and parallel to the coordinate system
used for A
2 1/30/14 Graphically Adding Vectors, cont.
•  Continue drawing the
vectors “tip-to-tail”
•  The resultant is drawn
from the origin of A to
the end of the last
vector
•  Measure the length of
and its angle
–  Use the scale factor to
convert length to actual
magnitude
Notes about Vector Addition
•  Vectors obey the
Commutative Law of
Addition
– The order in which
the vectors are
added does not
affect the result
–  A + B = B + A
Components of a Vector
•  A component is a
part
•  It is useful to use
rectangular
components
–  These are the
projections of the
vector along the xand y-axes
Graphically Adding Vectors, cont.
•  When you have
many vectors, just
keep repeating the
process until all are
included
•  The resultant is still
drawn from the
origin of the first
vector to the end of
the last vector
Vector Subtraction
•  Special case of
vector addition
–  Add the negative of
the subtracted vector
•  A - B = A + (-B)
•  Continue with
standard vector
addition procedure
Adding Vectors Algebraically
•  Choose a coordinate system and sketch
the vectors
•  Find the x- and y-components of all the
vectors
•  Add all the x-components
–  This gives Rx:
Rx = ∑ v x
3 1/30/14 Adding Vectors Algebraically, cont.
•  Add all the y-components
–  This gives Ry:
Ry = ∑ v y
•  Use the Pythagorean Theorem to find the
magnitude of the resultant:
•  Use the inverse tangent function to find
the direction of R:
θ = tan −1
Ry
Rx
Agenda
•  Today: Finish vectors L-T, projectile motion
& 2D problems (we’ll do circular motion
later)
•  HW #3 due on Tuesday
•  Start reading Chapter 4
R = R 2x + R 2y
Vector Addition Lecture-Tutorial
•  Work with a partner or two
•  Read directions and answer all questions
carefully. Take time to understand it now!
•  Come to a consensus answer you all agree
on before moving on to the next question.
•  If you get stuck, ask another group for help.
•  If you get really stuck, raise your hand and I
will come around.
Rules of Projectile Motion
Projectile Motion
•  An object may move in both the x and y
directions simultaneously
–  It moves in two dimensions
•  The form of two dimensional motion we will
deal with is called projectile motion
•  Assumptions:
–  We may ignore air friction
–  We may ignore the rotation of the earth
–  object in projectile motion will follow a parabolic
path
Projectile Motion
•  The x- and y-directions of motion are completely
independent of each other
•  The x-direction is uniform motion
–  ax = 0
•  The y-direction is free fall
–  ay = -g
•  The initial velocity can be broken down into its xand y-components
4 1/30/14 Projectile Motion at Various Initial
Angles
•  Complementary
values of the initial
angle result in the
same range
–  The heights will be
different
•  The maximum range
occurs at a
projection angle of
45o
More Details About the Rules
•  y-direction
–  free fall problem
•  a = -g
–  take the positive direction as upward
–  uniformly accelerated motion, so the motion
equations all hold
Some Details About the Rules
•  x-direction
–  ax = 0
–  vx is constant!
–  x = vxot
•  This is the only operative equation in the xdirection since there is uniform velocity in
that direction
Velocity of the Projectile
•  The velocity of the projectile at any point of
its motion is the vector sum of its x and y
components at that point
v = v x2 + vy2
and
θ = tan−1
vy
vx
–  Remember to be careful about the angle’s
quadrant
Problem-Solving Strategy
Problem-Solving Strategy, cont
•  Select a coordinate system and sketch the
path of the projectile
•  Follow the techniques for solving
problems with constant velocity to analyze
the horizontal motion of the projectile
•  Follow the techniques for solving
problems with constant acceleration to
analyze the vertical motion of the projectile
–  Include initial and final positions, velocities,
and accelerations
•  Resolve the initial velocity into x- and ycomponents
•  Treat the horizontal and vertical motions
independently
5 1/30/14 Some Variations of Projectile
Motion
•  An object may be
fired horizontally
•  The initial velocity is
all in the x-direction
–  vo = vx and vy = 0
•  All the general rules
of projectile motion
apply
Non-Symmetrical Projectile
Motion
•  Follow the general
rules for projectile
motion
•  Break the y-direction
into parts
–  up and down
–  symmetrical back to
initial height and then
the rest of the height
Projectile Motion Lecture-Tutorial
•  Work with a partner or two
•  Read directions and answer all questions
carefully. Take time to understand it now!
•  Come to a consensus answer you all agree
on before moving on to the next question.
•  If you get stuck, ask another group for help.
•  If you get really stuck, raise your hand and I
will come around.
Relative Velocity
•  Relative velocity is about relating the
measurements of two different observers
•  It may be useful to use a moving frame of
reference instead of a stationary one
•  It is important to specify the frame of reference,
since the motion may be different in different
frames of reference
•  There are no specific equations to learn to solve
relative velocity problems
Relative Velocity Notation
•  The pattern of subscripts can be useful in
solving relative velocity problems
•  Assume the following notation:
–  E is an observer (or the ground/Earth),
stationary with respect to the earth
–  A and B are two moving cars
6 1/30/14 Relative Position
•  The position of car A
relative to car B is
given by the vector
subtraction equation
•  This also works for
velocities!
vAE = vAE - vBE
Figure 3.18, p.54
Problem-Solving Strategy: Relative
Velocity
•  Label all the objects with a descriptive letter
•  Look for phrases such as “velocity of A
relative to B”
–  Write the velocity variables with appropriate
notation
–  If there is something not explicitly noted as being
relative to something else, it is probably relative
to the earth
Problem-Solving Strategy: Relative
Velocity, cont
•  Take the velocities and put them into an
equation
–  Keep the subscripts in an order analogous to
the standard equation
•  Solve for the unknown(s)
7