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:

• 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)

0

0

F

Some Shorthand: The Capital Delta• Whenever physicists

want the express how a quantity changes, they use a capital Greek Delta (Δ):

• Using the Delta we can define displacement, velocity and acceleration in a compact shorthand

(

t t t

x x x

f i

f i

quantity) (final quantity)-(initial quantity)

Examples: A change in time

final time - initial time (scalar)

final position - initial position (vector)

To express quantities which

x

t

change with time:

change in position which defines velocity.

change in time

Describing Motion:Four Basic Quantities

Quantity Scalar/

Vector

Symbol Definition Unit

Time scalar t Transition in Ce-133

Displacement vector Change in position

Velocity vector Change in position over some interval of time

Acceleration vector Change in velocity over some interval of time

f ix x x

xv

t

va

t

s

m

s

m

2s

m

• 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• An object rolling down an incline is still in

“free fall” under the influence of gravity!

at the same rate

Free Fall

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 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.

Galileo

Projectile Motion

• 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

Galileo

• Horizontal Motion(Constant Velocity)

Projectiles: Two Motions Superposed

g

• Vertical Motion (Constant Acceleration)

• The acceleration is g = 9.8 m/s2 everywhere along the trajectory, irrespective of position or velocity

CautionQuiz

Ahead

Concept Test #1A ball is thrown downward (not dropped) from the top of a tower. After being released, its downward acceleration will be:

A) Greater than g

B) Smaller than gC) Exactly g

The acceleration of gravity 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.

CautionQuiz

Ahead

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

g

• 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

Concept Test #3A 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.

A B A BH H t t

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

x

hTrajectory of dart without gravity

Trajectory of dart with gravity0v

θ

Both the bear and the dart are in “free fall”under the influence of gravity.

The dart hits the bear!

Nice trybear!

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!