Projectile Motion

1
Projectile Motion
Chapter
6.1
2
Projectile Motion
Section
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.
3
Projectile Motion
Section
6.1
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.

4
Projectile Motion
Section
6.1

• 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

5
Projectile Motion
Section
6.1
Independence of Motion in Two Dimensions
Click image to view movie. movanim 6.1
6
Projectile Motion
Section
6.1
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.

7
Projectile Motion
Section
6.1

• 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.

8
Projectile Motion
Section
6.1

• Vectors can also be used to represent a ball
  rolling horizontally on a table at a constant
  velocity.
• Newtons 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.

9
Projectile Motion
Section
6.1

• 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.

10
Projectile Motion
Section
6.1

• 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.

11
Projectile Motion
Section
6.1

• 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?

12
Section
Projectiles Launched at an Angle
6.1

• 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.

13
Section
Projectiles Launched at an Angle
6.1

• 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.

14
Section
Projectiles Launched at an Angle
6.1

• 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.

15
Section
Projectiles Launched at an Angle
6.1

• 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.

16
Section
Projectiles Launched at an Angle
6.1

• 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
17
Section
Projectiles Launched at an Angle
6.1

• Notice the x and y components of the velocity
  vector as the golf ball travels along its
  parabolic path.

18
Section
Projectiles Launched at an Angle
6.1

• 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).

19
Projectile Motion
Section
6.1

• 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.

20
Section Check
Section
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.)

1. The rocks fall with a constant vertical velocity
   and a constant horizontal acceleration.
2. The rocks fall with a constant vertical velocity
   as well as a constant horizontal velocity.
3. The rocks fall with a constant vertical
   acceleration and a constant horizontal velocity.
4. The rocks fall with a constant vertical
   acceleration and an increasing horizontal
   velocity.

21
Section Check
Section
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.
22
Section Check
Section
6.1
Question 2

• Which of the following conditions is met when a
  projectile reaches its maximum height?

1. Vertical component of the velocity is zero.
2. Vertical component of the velocity is maximum.
3. Horizontal component of the velocity is maximum.
4. Acceleration in the vertical direction is zero.

23
Section Check
Section
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.
24
Section Check
Section
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?

25
Section Check
Section
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.

26
Problem Solving with Projectile Motion

• Problem Solving Strategy
• Sketch the problem. List givens and unknowns.
• Divide the projectile motion into a vertical
  motion problem and a horizontal motion problem.
• 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.
• The horizontal motion of a projectile is the same
  as solving a constant velocity problem. Use dx
  vxt and vxi vxf.
• Vertical and horizontal motion are connected
  through
• the variable time .

27
Projectile Motion
Section
6.1

• Practice Problems, p. 150. 1 3.

HW 6.A
28
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
29
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 for a projectile that is launched
horizontally, g the time only
depends on the height Key Point In projectile
motion, the vertical and horizontal components
of motion are independent.