Transcript Motion in Two Dimensions - Derry Area School District

Chapter
6.1
Projectile Motion
Section
Projectile Motion
6.1
In this section you will:
Recognize that the vertical and horizontal motions of a
projectile are independent.
Relate the height, time in the air, and initial vertical velocity
of a projectile using its vertical motion, and then determine
the range using the horizontal motion.
Explain how the trajectory of a projectile depends upon the
frame of reference from which it is observed.
Read Chapter 6.1.
HW 6.A: Handout
Projectile Motion Study Guide, due before Chapter Test.
Section
6.1
Projectile Motion
Projectile Motion
If you observed the movement of a golf ball being hit from a tee,
a frog hopping, or a free throw being shot with a basketball, you
would notice that all of these objects move through the air along
similar paths, as do baseballs, arrows, and bullets.
Each path is a curve that moves upward for a distance, and
then, after a time, turns and moves downward for some
distance.
You may be familiar with this curve, called a parabola
from math class.
,
Section
6.1
Projectile Motion
An object shot through the air is called a projectile
.
A projectile can be a football, a bullet, or a drop of water.
You can draw a free-body diagram of a launched projectile and
identify all the forces that are acting on it.
No matter what the object is, after a projectile has been given an
initial thrust, if you ignore air resistance, it moves through the air
only under the force of gravity
The force of gravity is what causes the object to curve
downward in a parabolic flight path.
.
Its path through space is called its trajectory
.
Demonstration & Activity: Horizontal Projectile Motion
Section
6.1
Projectile Motion
Independence of Motion in Two Dimensions
Click image to view movie.
movanim 6.1
Section
6.1
Projectile Motion
Trajectories Depend upon the Viewer
The path of the projectile, or its trajectory, depends upon who is
viewing it.
Suppose you toss a ball up and catch it while riding in a bus. To
you, the ball would seem to go straight up and straight down.
But an observer on the sidewalk would see the ball leave your
hand, rise up, and return to your hand, but because the bus
would be moving, your hand also would be moving. The bus,
your hand, and the ball would all have the same horizontal
velocity.
Section
6.1
Projectile Motion
All objects, when ignoring air resistance, fall
with the same acceleration, g = 9.8 m/s2
downward.
The distance the ball falls each second
increases because the ball is accelerating
downward.
The velocity also increases in the downward
direction as the ball drops.
This is shown by drawing a longer vector
arrow for each time interval.
Section
6.1
Projectile Motion
Vectors can also be used to represent a ball rolling horizontally on
a table at a constant velocity.
Newton’s 1st Law tells us the ball will continue rolling in a straight
line at constant velocity unless acted on by an outside force.
Each vector arrow is drawn the same length to represent the
constant velocity. The velocity would remain constant but in the
real world, friction makes it slow down and eventually stop.
Section
6.1
Projectile Motion
Now, combine the motion of the ball in free fall with the motion of the ball
rolling on the table at a constant velocity.
This is seen when rolling the ball off of the table. The ball rolling on the
table would continue forever in a straight line if gravity is ignored. The ball
in free fall would continue to increase its speed if air resistance is ignored.
Section
6.1
Projectile Motion
Since the ball is moving at a constant velocity and in free fall at the same
time, the horizontal and vertical vectors are added together during equal
time intervals. This is done for each time interval until the ball hits the
ground.
The path the ball follows can be seen by connecting the resultant vectors.
Section
6.1
Projectile Motion
Look at the components
of the velocity vectors.
The length of the
horizontal component
stays the same for the
whole time.
The length of the vertical
component increases
with time.
How do we combine the
horizontal and vertical
components to find the
velocity vector?
Section
6.1
Projectiles Launched at an Angle
Demonstration: Tossing a Ball
If the object is launched upward, like a ball tossed straight up in
the air, it rises with slowing speed, reaches the top of its path,
and descends with increasing speed.
A projectile launched at an
angle would continue in a
straight line at a constant
velocity if gravity is
ignored. However, gravity
makes the projectile
accelerate to Earth. Notice
the projectile follows a
parabolic trajectory.
Section
6.1
Projectiles Launched at an Angle
Since the projectile is launched at an angle, it now has both horizontal
and vertical velocities.
The horizontal component of the velocity remains constant. The
vertical component of the velocity changes as the projectile moves up
or down.
Section
6.1
Projectiles Launched at an Angle
The up and right
vectors represent the
velocity given to the
projectile when
launched. The
vertical vectors
decrease in
magnitude due to
gravity. Eventually,
the effects of gravity
will reduce the upward
velocity to zero. This
occurs at the top of
the parabolic
trajectory where there
is only horizontal
motion.
Section
6.1
Projectiles Launched at an Angle
At the maximum height , the y component
of velocity is zero. The x component
remains constant.
After gravity reduces the
upward (vertical) speed
to zero it begins to add a
downward velocity. This
velocity increases until
the projectile return to
the ground.
Section
6.1
Projectiles Launched at an Angle
When looking at each half of the
trajectory (up and down) you can
determine that the speed of the projectile
going up is equal to the speed of the
projectile coming down (provided air
resistance is ignored). The only
difference is the direction of the motion.
The other quantity depicted is the
range
which is the horizontal
distance that the projectile travels.
Not shown is the flight time, which is
how much time the projectile is in the air.
For football punts, flight time often is
called hang time.
range
Section
6.1
Projectiles Launched at an Angle
Notice the x and y components of the velocity vector as the golf ball
travels along its parabolic path.
Section
6.1
Projectiles Launched at an Angle
Maximum range is achieved with a projection angle of 45°
.
For projection angles above and below 45°, the range is shorter, and
it is equal for angles equally different from 45° (for example, 30° and
60°).
Section
6.1
Projectile Motion
So far, air resistance has been ignored in the analysis of projectile
motion.
While the effects of air resistance are very small for some projectiles,
for others, the effects are large and complex. For example, dimples
on a golf ball reduce air resistance and maximize its range.
The force due to air resistance does exist and it can be important.
Section
Section Check
6.1
Question 1
A boy standing on a balcony drops a rock and throws another with
an initial horizontal velocity of 3 m/s. Which of the following
statements about the horizontal and vertical motions of the rocks are
correct? (Neglect air resistance.)
A. The rocks fall with a constant vertical velocity and a constant
horizontal acceleration.
B. The rocks fall with a constant vertical velocity as well as a
constant horizontal velocity.
C. The rocks fall with a constant vertical acceleration and a
constant horizontal velocity.
D. The rocks fall with a constant vertical acceleration and an
increasing horizontal velocity.
Section
Section Check
6.1
Answer 1
Answer: C
Reason: The vertical and horizontal motions of a projectile are
independent. The only force acting on the two rocks is
force due to gravity. Because it acts in the vertical
direction, the balls accelerate in the vertical direction. The
horizontal velocity remains constant throughout the flight of
the rocks.
Section
Section Check
6.1
Question 2
Which of the following conditions is met when a projectile reaches its
maximum height?
A. Vertical component of the velocity is zero.
B. Vertical component of the velocity is maximum.
C. Horizontal component of the velocity is maximum.
D. Acceleration in the vertical direction is zero.
Section
Section Check
6.1
Answer 2
Answer: A
Reason: The maximum height is the height at which the object
stops its upward motion and starts falling down, i.e. when
the vertical component of the velocity becomes zero.
Section
Section Check
6.1
Question 3
Suppose you toss a ball up and catch it while riding in a bus. Why
does the ball fall in your hands rather than falling at the place where
you tossed it?
Section
Section Check
6.1
Answer 3
Trajectory depends on the frame of reference.
For an observer on the ground, when the bus is moving, your hand
is also moving with the same velocity as the bus, i.e. the bus, your
hand, and the ball will have the same horizontal velocity. Therefore,
the ball will follow a trajectory and fall back in your hands.
Section
6.1
Problem Solving with Projectile Motion
Problem Solving Strategy
1. Sketch the problem. List givens and unknowns.
2. Divide the projectile motion into a vertical motion problem and a
horizontal motion problem.
3. The vertical motion of a projectile is exactly that of an object dropped
or thrown straight up or down with constant acceleration g. Use your
constant acceleration (kinematics) equations.
4. The horizontal motion of a projectile is the same as solving a
constant velocity problem. Use dx = vxt and vxi = vxf.
5. Vertical and horizontal motion are connected through
the variable time
.
Section
Projectile Motion
6.1
Practice Problems, p. 150. 1 – 3.
HW 6.A
Section
6.1
Physics Chapter 6 Test Information
The test is worth 45 points total.
Multiple Choice: 7 questions, 1 point each
Problem Solving: 28 points
Short Answer: 10 points
Section
6.1
Projectile Motion Review
Formulas:
dy = vit + ½ a t2
constant acceleration in the y- direction
dx = vx t
constant velocity in the x- direction
t = 2dy
g
for a projectile that is launched horizontally,
the time only depends on the height
Key Point: In projectile motion, the vertical and
horizontal components of motion are
independent.