Transcript Projectiles & CF

Projectile Motion and
Centripetal Force
Projectile Motion
Projectile motion: motion that travels
along a curved path.
 This curve is a combination of horizontal
motion (motion along the x-axis) and vertical
motion (motion along the y-axis).
 Ex: A ball thrown thru the air, a cannonball
launched
Projectile: any object that moves
through the air or space, acted on
only by gravity (and air resistance,
if any)
 Accelerating in the vertical direction due to
the force of gravity
 Moving at a constant velocity in the
horizontal direction due to inertia
 Ex: sports ball, object in free fall, bullet
• The horizontal motion (motion along the xaxis) of a projectile is at a constant velocity
because there is NO force in the horizontal
direction.
 It keeps moving in the x-axis direction
because of inertia!
• The vertical motion of a projectile is at
constant acceleration because gravity is
ALWAYS acting on it.
 Gravity slows the upward motion, and gravity
speeds up the downward motion of the
projectile.
• THE HORIZONTAL MOTION FOR A
PROJECTILE IS COMPLETELY
INDEPENDENT OF THE VERTICLE
COMPONENT OF MOTION!!!!!!!!!!!!
 The combined effects is what produces the
curved paths that projectile follow.
Check Your Understanding
At the instant a horizontally pointed
cannon is fired, a cannonball held at
the cannon’s side is released and drops
to the ground. Which cannonball
strikes the ground first, the one fired
from the cannon or the one dropped?
Both cannonballs fall the same vertical
distance with the same acceleration due to
gravity; therefore they strike the ground at
the same time. (Remember that horizontal
velocity is independent of vertical velocity).
Range of Projectiles
• Horizontal range
 Projectiles will have a maximum horizontal range if
launched at a 45 degree angle
 As the angle moves away from 45 degrees, the
horizontal range decreases
 89 and 1 degrees will have the shorter ranges
 44 and 46 will have longer ranges
• Vertical range
 The steeper the angle at which a projectile is
launched, the higher the path will be.
 Projectiles launched at steep angles do not travel far
in the horizontal direction.
• When air resistance is significant, the range of
the projectile is below its ideal.
Check Your Understanding
A projectile is launched into the air.
Neglecting air resistance, what is its
vertical acceleration? Its horizontal
acceleration?
Its vertical acceleration is g (10 m/s2)
because the force of gravity is downward.
Its horizontal acceleration is zero because
no horizontal force acts on it.
Check Your Understanding
At what point in its path does a projectile
have minimum speed?
The minimum speed of a projectile occurs
at the top of its path. The speed at the top
is the horizontal velocity as there will be no
vertical velocity at the peak.
Rotation & Revolution
Axis: a straight line around which
rotation takes place
 Objects spin on their axis
 Ex: the North-South pole line thru the earth
Rotation: axis is located within the
body of the object
 An object spinning about on its axis
 Ex: spinning skater
Revolution: when an object turns
about an external axis
 Object moves AROUND another object
 Ex: earth revolves around the sun in 365 days
Rotation vs. Revolution
• The turntable rotates about its axis.
• The lady bug revolves around the same axis
Check Your Understanding
Does a tossed football rotate or revolve?
 rotates about its own axis
Does a ball whirled overhead at the end
of a string rotate or revolve?
 it revolves around you
Rotational Speed vs Linear Speed
Linear Speed: distance moved per
unit of time
 varies with the distance moved from the axis
 the further away from the axis, the greater the
linear speed
Rotational Speed: number of
rotations per unit time
 does NOT vary with distance from the axis
Rotational Speed vs. Linear Speed
• All parts of the turntable rotate at the same
speed, but the ladybugs at different distances
from the center travel at different linear speeds.
• A ladybug sitting twice as far from the center
moves twice as fast.
Check Your Understanding
Which part of Earth’s surface has the
greatest rotational speed relative to
Earth’s axis?
all parts of Earth have the same rotational
speed.
Which part of Earth’s surface has the
greatest linear speed relative to Earth’s
surface?
The equator has the greatest linear speed
because it is furthest from the axis.
Check Your Understanding
If a meter stick supported at the 0cm
mark swings like a pendulum from your
fingers, how fast at any given moment
is the 100cm mark moving compared to
the 50cm mark?
Twice as fast for linear speed because the
100 cm mark is twice as far from the axis
of rotation. The rotational speed is the
same everywhere.
Centripetal Force
Centripetal force: any force that
causes an object to follow a circular path
 Centripetal means center seeking or towards
the center
 SI Unit: Newtons (N)
 Equation: Fc = mv2 / r
•
•
•
•
Fc = centripetal force (N)
m = mass (kg)
v = velocity (m/s)
r = radius (m)
• Mass and centripetal force are directly
proportional




2m = 2FC
½m = ½FC
10m = 10FC
1/10m = 1/10FC
• Radius and centripetal force are inversely
proportional




2r = ½FC
½r = 2FC
10r = 1/10FC
1/10r = 10FC
• As an object moves faster around a curve,
the velocity is directly squared to the
centripetal force
 2v = 22 or 4F
 3v = 32 or 9F
 10v = 102 or 100F
**This is why you must decrease your speed
considerably when going around a curve in
your car**
Examples of Centripetal Forces
• As a car makes a
turn, the force of
friction acting upon
the turned wheels of
the car provides the
centripetal force
required to keep the
car in circular motion.
• As a bucket of water
is tied to a string and
spun in a circle, the
force of tension acting
upon the bucket
provides the
centripetal force
required for circular
motion
• As the moon orbits
Earth, the force of
gravity acting upon
the moon provides
the centripetal force
required for circular
motion.
Check Your Understanding
A motorcycle runs on the inside of a
bowl-shaped track. Is the force that
holds the motorcycle in a circular path
an inward- or outward- directed force?
It is an inward-directed force – also known
as a centripetal force.
• When the string breaks, the whirling can moves
in a straight line, tangent to its circular path.
Notice that the can does not move outward from
the center
!!!!Important !!!!
• In order for circular motion to take place, a
centripetal force must be present!
• The reason that objects are “pushed” to the
outside of a circle is because they have inertia!
– Ex: your clothes being pushed to the outside of the
washing machine during the spin cycle; your behind
remaining in the seat on a loop-de-loop on a roller
coaster.
Check Your Understanding
A tin can, with a mass of 1 kg, is on a
string with a length of 2 m. If the can is
being whirled around someone’s head
at 4 m/s, what is the centripetal force
acting on the can?
 Fc = ?
m = 1 kg
v = 4 m/s
r=2m
Fc = mv2 / r
Fc = (1)(4)(4) / (2) = 8 Newtons
Check Your Understanding
If the string on a tin can suddenly breaks
as it is being whirled overhead, which
way does it fly?
In a straight line tangent to the circle.
Why?
It’s inertia will allow it to move in the
direction that it was already moving in (in
non-accelerated motion…not turning or
changing speed).