Download projectile

Focus Question(s)
 If you shoot a dart at a low angle (0), then
continue to shoot at a greater and greater
angle (90), how will that effect…
 the distance the dart travels?
 the time the dart spends in the air?
Chapter 6
Motion in Two Dimensions
6.1 - Projectile Motion
 A projectile is an object that is shot through
the air.
Three types of projectiles:
 Projectiles shot vertically (90 degrees)
 Projectiles shot horizontally (0 degrees)
 Projectiles shot at some angle between 0-90 degrees
Projectiles
 The path that a projectile takes while in
flight is called the trajectory. The shape of
this path is called a parabolic curve (or
parabola).
 The horizontal distance that the projectile
travels is called its final displacement.
 The highest point in the trajectory is called
the max height.
Hangtime
 The time a projectile spends in the air is called its
hangtime.
 If shot horizontally, the time is only the time
down.
 If shot at any angle greater than 0 degrees, the
projectile will travel up for a while, then down.
The time up is equal to the time down.
 The closer the launch angle is to 90 degrees, the
more time the projectile will spend in the air
(assuming that the initial velocity is constant).
Diagram of Horizontal Projectile
vx
vx= horizontal velocity
dy
dy= vertical height
t = time to fall
t
dx= final displacement
dx
Diagram of an Angled Projectile
dy-max
Time up equals time down
dx
Horizontal Projectile formulas
 The horizontal velocity of a projectile does
not change while in flight. Same formula
from Chapter 2:
dx
vx 
t
Horizontal Projectile formulas
 To find the time a horizontal projectile stays
in the air, you only need to know how high
it was dropped from (the only acceleration
for a projectile is that of gravity):
2dy 
t   
 g 
Horizontal Projectile Formulas
 To find the vertical velocity of a projectile
that was initially shot horizontally, you will
need to know the time that it was in the air:
vy f  vyi  at
Example 1
 A ball is thrown from the top of a 125m tall
structure with an initial horizontal velocity
of 15m/s.
 How long will it take for it to hit the ground?
 How far will it land from the base of the
structure?
 What was the vertical component of the
velocity immediately before the ball hit the
ground?
Example 2
 An arrow is shot horizontally from 2.0m
above the ground and lands 75m away from
the shooter.
 How long was it in the air?
 What was the initial speed of the arrow?
Homework
 P. 150, #1-3
Journal #32
10/23/08
 A bullet is dropped from 2.0m at the exact
same time that a bullet is shot horizontally
from 2.0m. Which bullet will hit the ground
first? Why?
Journal #32 Answer
 Velocity is not part of
the time formula, so
it has no effect on
time.
2dy 
t   
 g 
Answers to HW
Answers to HW
6.2 - Circular Motion
 There are many objects that do not travel in
straight lines or move along a trajectory.
 Some objects travel in a circular motion:
 Blades of a fan
 Cars going around a curve
 Satellites orbiting the earth
Uniform Circular Motion
 The movement of an object or particle
trajectory at a constant speed around a circle
with a fixed radius (r).
 As an object moves around the circle, the
length of the radius does not change.
 The acceleration of the object is toward the
center of the circle causing the velocity to
stay at a tangent. A tangent line is a line
that passes through a single point of a circle
and is perpendicular to the radius of that
circle.
Diagram of Circular Motion
Centripetal Acceleration
 Centripetal acceleration always points to
the center of the circle. Its magnitude is
equal to the square of the velocity, divided
by the radius of circular motion.
2
v
ac 
r
How can you measure the speed of an object
moving in a circle?
 Speed is still calculated as distance traveled divided by the
time it takes to travel that distance.
 In the case of circular motion, the distance is the
circumference of a circle or (2πr).
 The time it takes to go around a circle one time is called
the period (T).
(2r)
v
T
The “New” Net Force?
We know that an object must have a force
acting on it for it to accelerate… so what is
this mysterious force that is pulling or
pushing towards the center of the circle?
 A net force that in a direction towards the
center of a circle is called the centripetal
force.
Centripetal Force
 The centripetal force is really a costume
artist… it can be whichever force is causing
the acceleration to the center of the circular
motion.
 Examples:
 A car going around a curve - Friction Force
 A satellite in orbit - Force of Gravity
 A ball on a string - Tension Force
 The first step for any circular motion
problem is to identify the force that is acting
as the centripetal force.
Using Newton’s Second Law Again
 Newton’s 2nd Law stated that net force on
an object is equal to the mass times the
acceleration.
 This holds true, but we substitute in
centripetal acceleration:
Fnet  mac
Does Centrifugal Force Exist???
 If you have ever been in a car that suddenly
turned to the left, your body may have been
thrown to the right… does that mean that
there is some outward force?
 No… the car around you has moved and
you have tried to maintain your original
path (according to Newton’s 1st Law).
Example 1
 An athlete whirls a 7.00kg hammer 1.8m
from the axis of rotation in a horizontal
circle. The hammer makes one revolution
in 1.0s.
 What is the centripetal acceleration of the
hammer?
 What is the tension in the chain?
Example 2
 If a 0.40kg stone is whirled horizontally on
the end of a 0.60m string at a speed of
2.2m/s, what is the tension in the string?
HW: P. 156, #12-15