Introduction to 2-Dimensional Motion

1
Introduction to 2-Dimensional Motion
2
2-Dimensional Motion

• Definition motion that occurs with both x and y
  components.
• Example
• Playing pool .
• Throwing a ball to another person.
• Each dimension of the motion can obey different
  equations of motion.

3
Solving 2-D Problems

• Resolve all vectors into components
• x-component
• Y-component
• Work the problem as two one-dimensional problems.
• Each dimension can obey different equations of
  motion.
• Re-combine the results for the two components at
  the end of the problem.

4
Sample Problem

• You run in a straight line at a speed of 5.0 m/s
  in a direction that is 40o south of west.
• How far west have you traveled in 2.5 minutes?
• How far south have you traveled in 2.5 minutes?

5
Sample Problem

• You run in a straight line at a speed of 5.0 m/s
  in a direction that is 40o south of west.
• How far west have you traveled in 2.5 minutes?
• How far south have you traveled in 2.5 minutes?

v 40 m/s
6
Sample Problem

• You run in a straight line at a speed of 5.0 m/s
  in a direction that is 40o south of west.
• How far west have you traveled in 2.5 minutes?
• How far south have you traveled in 2.5 minutes?
• v 5 m/s, Q 40o,
• t 2.5 min 150 s
• vx v cosQ vy v sin Q
• vx 5 cos 40 vy 5 sin 40
• vx vy
• x vx t y vyt
• x ( )(150) y ( )(150)
• x y

vx
vy
v 5 m/s
7
Sample Problem

• A particle passes through the origin with a speed
  of 6.2 m/s traveling along the y axis. If the
  particle accelerates in the negative x direction
  at 4.4 m/s2.
• What are the x and y positions at 5.0 seconds?

8
Sample Problem

• A particle passes through the origin with a speed
  of 6.2 m/s traveling along the y axis. If the
  particle accelerates in the negative x direction
  at 4.4 m/s2.
• What are the x and y positions at 5.0 seconds?
• Vo,y 6.2 m/s, Vo,x 0 m/s, t 5 s, ax
  -4.4 m/s2, ay 0 m/s2
• x ? y ?
• x vo,x at y vo,y at
• x 0 (-4.4)(5) y 6.2 (0)(5)
• x y

9
Sample Problem

• A particle passes through the origin with a speed
  of 6.2 m/s traveling along the y axis. If the
  particle accelerates in the negative x direction
  at 4.4 m/s2.
• What are the x and y components of velocity at
  this time?

10
Sample Problem

• A particle passes through the origin with a speed
  of 6.2 m/s traveling along the y axis. If the
  particle accelerates in the negative x direction
  at 4.4 m/s2.
• What are the x and y components of velocity at
  this time?
• vx vo,x axt vy voy axt
• vx 0 (-4.4)(5) vy 6.2 0(5)
• vx vy

11
Projectiles
12
Projectile Motion

• Something is fired, thrown, shot, or hurled near
  the earths surface.
• Horizontal velocity is constant.
• Vertical velocity is accelerated.
• Air resistance is ignored.

13
1-Dimensional Projectile

• Definition A projectile that moves in a vertical
  direction only, subject to acceleration by
  gravity.
• Examples
• Drop something off a cliff.
• Throw something straight up and catch it.
• You calculate vertical motion only.
• The motion has no horizontal component.

14
2-Dimensional Projectile

• Definition A projectile that moves both
  horizontally and vertically, subject to
  acceleration by gravity in vertical direction.
• Examples
• Throw a softball to someone else.
• Fire a cannon horizontally off a cliff.
• Shoot a monkey with a blowgun.
• You calculate vertical and horizontal motion.

15
Horizontal Component of Velocity

• Is constant
• Not accelerated
• Not influence by gravity
• Follows equation
• x Vo,xt

16
Horizontal Component of Velocity
17
Vertical Component of Velocity

• Undergoes accelerated motion
• Accelerated by gravity (9.8 m/s2 down)
• Vy Vo,y - gt
• y yo Vo,yt - 1/2gt2
• Vy2 Vo,y2 - 2g(y yo)

18
Horizontal and Vertical
19
Horizontal and Vertical
20
Zero Launch Angle Projectiles
21
Launch angle

• Definition The angle at which a projectile is
  launched.
• The launch angle determines what the trajectory
  of the projectile will be.
• Launch angles can range from -90o (throwing
  something straight down) to 90o (throwing
  something straight up) and everything in between.

22
Zero Launch angle

• A zero launch angle implies a perfectly
  horizontal launch.

23
Sample Problem

• The Zambezi River flows over Victoria Falls in
  Africa. The falls are approximately 108 m high.
  If the river is flowing horizontally at 3.6 m/s
  just before going over the falls, what is the
  speed of the water when it hits the bottom?
  Assume the water is in freefall as it drops.

24
Sample Problem

• The Zambezi River flows over Victoria Falls in
  Africa. The falls are approximately 108 m high.
  If the river is flowing horizontally at 3.6 m/s
  just before going over the falls, what is the
  speed of the water when it hits the bottom?
  Assume the water is in freefall as it drops.
• yo 108 m, y 0 m, g -9.8 m/s2, vo,x 3.6
  m/s
• v ?

25
Sample Problem

• The Zambezi River flows over Victoria Falls in
  Africa. The falls are approximately 108 m high.
  If the river is flowing horizontally at 3.6 m/s
  just before going over the falls, what is the
  speed of the water when it hits the bottom?
  Assume the water is in freefall as it drops.
• yo 108 m, y 0 m, g 9.8 m/s2, vo,x 3.6
  m/s
• v ?
• Gravity doesnt change horizontal
  velocity. vo,x vx 3.6 m/s
• Vy2 Vo,y2 - 2g(y yo)
• Vy2 (0)2 2(9.8)(0 108)
• Vy

26
Sample Problem

• The Zambezi River flows over Victoria Falls in
  Africa. The falls are approximately 108 m high.
  If the river is flowing horizontally at 3.6 m/s
  just before going over the falls, what is the
  speed of the water when it hits the bottom?
  Assume the water is in freefall as it drops.
• yo 108 m, y 0 m, g 9.8 m/s2, vo,x 3.6
  m/s
• v ?
• Gravity doesnt change horizontal
  velocity. vo,x vx 3.6 m/s
• Vy2 Vo,y2 - 2g(y yo)
• v Vy2 (0)2 2(9.8)(0 108)
• Vy

27
Sample Problem

• An astronaut on the planet Zircon tosses a rock
  horizontally with a speed of 6.75 m/s. The rock
  falls a distance of 1.20 m and lands a horizontal
  distance of 8.95 m from the astronaut. What is
  the acceleration due to gravity on Zircon?

28
Sample Problem

• An astronaut on the planet Zircon tosses a rock
  horizontally with a speed of 6.75 m/s. The rock
  falls a distance of 1.20 m and lands a horizontal
  distance of 8.95 m from the astronaut. What is
  the acceleration due to gravity on Zircon?
• vo,x 6.75 m/s, x 8.95 m, y 0 m, yo 1.2,
  Vo,y 0 m/s
• g ?
• y yo Vo,yt - 1/2gt2
• g -2(y - yo - Vo,yt)/t2
• g -20 1.2 (0)( )/( )2
• g

x vo,xt t x/vo,x t 8.95/6.75 t
29
Sample Problem

• Playing shortstop, you throw a ball horizontally
  to the second baseman with a speed of 22 m/s. The
  ball is caught by the second baseman 0.45 s
  later.
• How far were you from the second baseman?
• What is the distance of the vertical drop?
• Should be able to do this on your own!

30
General Launch Angle Projectiles
31
General launch angle

• Projectile motion is more complicated when the
  launch angle is not straight up or down (90o or
  90o), or perfectly horizontal (0o).

32
General launch angle

• You must begin problems like this by resolving
  the velocity vector into its components.

33
Resolving the velocity

• Use speed and the launch angle to find horizontal
  and vertical velocity components

Vo
?
34
Resolving the velocity

• Then proceed to work problems just like you did
  with the zero launch angle problems.

Vo
?
35
Sample problem

• A soccer ball is kicked with a speed of 9.50 m/s
  at an angle of 25o above the horizontal. If the
  ball lands at the same level from which is was
  kicked, how long was it in the air?

36
Sample problem

• A soccer ball is kicked with a speed of 9.50 m/s
  at an angle of 25o above the horizontal. If the
  ball lands at the same level from which is was
  kicked, how long was it in the air?
• vo 9.5 m/s, q 25o, g -9.8 m/s2, Remember
  because it lands at the same height Dy y yo
  0 m and vy - vo,y
• Find Vo,y Vo sin ? and Vo,x Vo cos ?
• Vo,y 9.5 sin 25 Vo,x Vo cos ?
• Vo,y Vo,x
• t ?
• Vy Vo,y - gt
• t (Vy - Vo,y )/g
• t ( ) ( )/9.8 dont forget vy -
  vo,y
• t

37
Sample problem

• Snowballs are thrown with a speed of 13 m/s from
  a roof 7.0 m above the ground. Snowball A is
  thrown straight downward snowball B is thrown in
  a direction 25o above the horizontal. When the
  snowballs land, is the speed of A greater than,
  less than, or the same speed of B? Verify your
  answer by calculation of the landing speed of
  both snowballs.
• Well do this in class.

38
Projectiles launched over level ground

• These projectiles have highly symmetric
  characteristics of motion.
• It is handy to know these characteristics, since
  a knowledge of the symmetry can help in working
  problems and predicting the motion.
• Lets take a look at projectiles launched over
  level ground.

39
Trajectory of a 2-D Projectile

• Definition The trajectory is the path traveled
  by any projectile. It is plotted on an x-y graph.

40
Trajectory of a 2-D Projectile

• Mathematically, the path is defined by a parabola.

41
Trajectory of a 2-D Projectile

• For a projectile launched over level ground, the
  symmetry is apparent.

42
Range of a 2-D Projectile
Range

• Definition The RANGE of the projectile is how
  far it travels horizontally.

43
Maximum height of a projectile
Maximum Height
Range

• The MAXIMUM HEIGHT of the projectile occurs when
  it stops moving upward.

44
Maximum height of a projectile
Maximum Height
Range

• The vertical velocity component is zero at
  maximum height.

45
Maximum height of a projectile
Maximum Height
Range

• For a projectile launched over level ground, the
  maximum height occurs halfway through the flight
  of the projectile.

46
Acceleration of a projectile

• Acceleration points down at 9.8 m/s2 for the
  entire trajectory of all projectiles.

47
Velocity of a projectile
v
v
v
vo
vf

• Velocity is tangent to the path for the entire
  trajectory.

48
Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• The velocity can be resolved into components all
  along its path.

49
Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• Notice how the vertical velocity changes while
  the horizontal velocity remains constant.

50
Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• Maximum speed is attained at the beginning, and
  again at the end, of the trajectory if the
  projectile is launched over level ground.

51
Velocity of a projectile

• Launch angle is symmetric with landing angle for
  a projectile launched over level ground.

52
Time of flight for a projectile

• The projectile spends half its time traveling
  upward

53
Time of flight for a projectile

• and the other half traveling down.

54
Position graphs for 2-D projectiles
55
Velocity graphs for 2-D projectiles
Vy
Vx
t
t
56
Acceleration graphs for 2-D projectiles
ay
ax
t
t
57
Projectile Lab
58
Projectile Lab

• The purpose is to collect data to plot a
  trajectory for a projectile launched
  horizontally, and to calculate the launch
  velocity of the projectile. Equipment is
  provided, you figure out how to use it.
• What you turn in
• a table of data
• a graph of the trajectory
• a calculation of the launch velocity of the ball
  obtained from the data
• Hints and tips
• The thin paper strip is pressure sensitive.
  Striking the paper produces a mark.
• You might like to hang a sheet of your own graph
  paper on the brown board.

59
More on Projectile Motion
60
The Range Equation

• Derivation is an important part of physics.
• Your book has many more equations than your
  formula sheet.
• The Range Equation is in your textbook, but not
  on your formula sheet. You can use it if you can
  memorize it or derive it!

61
The Range Equation

• R vo2sin(2q)/g.
• R range of projectile fired over level ground
• vo initial velocity
• g acceleration due to gravity
• q launch angle

62
Deriving the Range Equation
63
Sample problem

• A golfer tees off on level ground, giving the
  ball an initial speed of 42.0 m/s and an initial
  direction of 35o above the horizontal.
• How far from the golfer does the ball land?
• Vo 42m/s, q 35o, g 9.8 m/s2
• R ?

64
Sample problem

• A golfer tees off on level ground, giving the
  ball an initial speed of 42.0 m/s and an initial
  direction of 35o above the horizontal.
• The next golfer hits a ball with the same initial
  speed, but at a greater angle than 45o. The ball
  travels the same horizontal distance. What was
  the initial direction of motion?