Physics 106 Lesson #3
Motion Under the Influence
of Gravity: Galileo’s
Experiment and Projectile
Motion
Dr. Andrew Tomasch
2405 Randall Lab
atomasch@umich.edu
Last time: Equilibrium for Rigid Bodies
• Extended objects
which do not change
shape are called rigid
bodies
• For a rigid body to be
in equilibrium:
F

0



0

• For a rigid body to
remain stationary in one
place, the net force
acting on it must be zero
• For a rigid body to
remain stationary and
not rotate the net torque
about any axis through
the body must be zero
• The net force/torque is
the vector sum of all
forces/torques acting on
the body (analyze with
Free Body Diagram)
Some Shorthand: The Capital Delta
• Whenever physicists
• Using the Delta we can
want the express how a define displacement,
quantity changes, they
velocity and
use a capital Greek
acceleration in a
Delta (Δ):
compact shorthand
 (quantity)  (final quantity)-(initial quantity)
Examples: A change in time 
t  tf  ti  final time - initial time (scalar)
x  xf  xi  final position - initial position (vector)
To express quantities which change with time:
x change in position

which defines velocity.
t
change in time
Describing Motion:
Four Basic Quantities
Quantity
Time
Scalar/ Symbol Definition
Unit
Vector
s
scalar t
Transition in Ce-133
Displacement vector
x  xf  xi Change in position
m
Velocity
vector
x Change in position
v
over some interval
t of time
m
s
Acceleration
vector
v Change in velocity
a
over some interval
t of time
m
s2
Free Fall
• What is “free fall”?
– refers to objects that are dropped (or thrown)
vertically
• The earth’s gravity produces a nearly
constant acceleration of g = 9.8 m/s2
downward
• An object dropped from rest reaches a
velocity of 9.8 m/s (~24 mph) after one
second
• In vacuum all objects accelerate at the same rate
• An object rolling down an incline is still in
“free fall” under the influence of gravity!
Demonstration:
The Feather and the Farthing
• All objects near the earth are attracted
with the same acceleration g = 9.8 m/s2
toward the Earth’s center (“downward”).
• In a vacuum all objects will fall with the
same acceleration when released from
rest, independent of their masses or
Galileo
sizes.
• Objects fall with different accelerations in
air only because of the drag force (“air
resistance”) produced by the air on the
falling object
• The feather and the farthing will fall
together in a vacuum.
Projectile Motion
Galileo
• Galileo: the vertical and horizontal motions
of a projectile are independent
• The horizontal motion is motion with
constant velocity
• The vertical motion is motion with constant
acceleration ( g the acceleration of gravity)
• The horizontal motion and the vertical
motion do not affect each other
Projectiles: Two Motions Superposed
• Horizontal Motion
(Constant Velocity)
g
q
• Vertical Motion
(Constant
Acceleration)
• The acceleration is
g = 9.8 m/s2 everywhere
along the trajectory,
irrespective of position
or velocity
Caution
Quiz
Ahead
Concept Test #1
Caution
Quiz
Ahead
A ball is thrown downward (not dropped)
from the top of a tower. After being
released, its downward acceleration will be:
A) Greater than g The acceleration of gravity
B) Smaller than g
C) Exactly g
always points downward
with g = 9.8 m/s2 near the
surface of the Earth. Once
the ball is released it is in
free fall with g = 9.8 m/s2
downward.
Concept Test #2
In a baseball game, a batter hits a pitched ball that
goes straight up in the air above home plate. At the
top of its motion, the ball is momentarily
motionless. At this point
A) the acceleration is zero because the ball is
motionless at that instant.
B) the acceleration is zero because the motion is
changing from slowing down to speeding up.
C) the acceleration is zero because at that
instant the force from the impact with the bat
is balanced by the pull of the earth.
D) the acceleration is 9.8 m/s2 downward.
Projectiles: Key Points
• The vertical and horizontal
motions are independent
and do not affect each
other
• The time-of-flight tof
depends only on the
maximum height H and
increases with H
• The range R is the product
of tof and the horizontal
component of velocity at
launch
• The longest rage for a
given launch speed
occurs at a launch angle
of 45 degrees
g
q
Concept Test #3
A battle ship simultaneously fires two shells at
enemy ships. If the shells follow the parabolic
trajectories shown, which ship gets hit first?
A) Ship A.
B) Ship B.
C) They will both be hit at the same time.
D) Need more information.
H A  H B  tA  tB
Demonstration: The Slug Gun
A slug gun is a device which fires a
billiard ball exactly horizontally while
dropping another vertically. Which ball
will hit the ground first?
The horizontal velocity does not
affect how an object falls
vertically. Therefore, both balls
hit the ground at the same time.
Shooting the Falling Bear
y
h
Trajectory of dart
without gravity
v0
θ
Trajectory of dart
with gravity
Both the bear and the dart are in “free fall”
under the influence of gravity.
The dart hits the bear!
Nice try
bear!
x
Galileo’s Last Word: A Puzzle
• All objects near the earth are
attracted with the same
acceleration g = 9.8 m/s2 toward the
Earth’s center (“downward”)
irrespective of their masses
• Forces cause motion
• More massive objects have a
larger downward force acting on
them due to gravity
• How can these facts be
consistent?
• Enter Sir Isaac Newton!