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Kinematics in2-Dimensional Motions
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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.
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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.
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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.a) How far west have you traveled in 2.5 minutes?b) How far south have you traveled in 2.5 minutes?
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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.a) How far west have you traveled in 2.5 minutes?b) How far south have you traveled in 2.5 minutes?
40o
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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.
a) How far west have you traveled in 2.5 minutes?b) How far south have you traveled in 2.5 minutes?
v = 5 m/s, = 40o, t = 2.5 min = 150 svx = v cosvy= v sin
vx = 5 cos 40vy= 5 sin 40
vx = vy=
x = vx t y = vyt
x = ( )(150) y = ( )(150)x = y =
v = 5 m/s
vx
vy
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Projectiles
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Projectile Motion Something is fired, thrown, shot, or
hurled near the earth’s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored.
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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.
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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.
You calculate vertical and horizontal motion.
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Horizontal Component of Velocity Is constant Not accelerated Not influence by gravity Follows equation: x = Vo,xt
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Horizontal Component of Velocity
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Vertical Component of Velocity Undergoes accelerated motion Accelerated by gravity (9.81 m/s2
down) Vy = Vo,y - gt y = yo + Vo,yt - 1/2gt2
Vy2 = Vo,y
2 - 2g(y – yo)
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Horizontal and Vertical
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Zero Launch Angle Projectiles
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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.
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Zero Launch angle
A zero launch angle implies a perfectly horizontal launch.
vo
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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.
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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 = -9.81 m/s2, vo,x = 3.6 m/s
v = ?
2y
2x vvv
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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 doesn’t change horizontal velocity. vo,x = vx =
3.6 m/s
Vy2 = Vo,y
2 - 2g(y – yo)
Vy2 = (0)2 – 2(9.8)(0 – 108)
Vy =
2y
2x vvv
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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 doesn’t change horizontal velocity. vo,x = vx =
3.6 m/s
Vy2 = Vo,y
2 - 2g(y – yo)
v = Vy2 = (0)2 – 2(9.8)(0 – 108)
Vy =
2y
2x vvv
22 ) ((3.6)v
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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.
a) How far were you from the second baseman?b) What is the distance of the vertical drop?
Should be able to do this on your own!
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General Launch Angle Projectiles
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General launch angle
vo
Projectile motion is more complicated when the launch angle is not straight up or down (90o or –90o), or perfectly horizontal (0o).
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General launch angle
vo
You must begin problems like this by resolving the velocity vector into its components.
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Resolving the velocity Use speed and the launch angle to find
horizontal and vertical velocity components
VoVo,y = Vo sin
Vo,x = Vo cos
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Resolving the velocity Then proceed to work problems just like
you did with the zero launch angle problems.
VoVo,y = Vo sin
Vo,x = Vo cos
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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?
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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?
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To Be Continued…
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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.
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Trajectory of a 2-D Projectile
x
y
Definition: The trajectory is the path traveled by any projectile. It is plotted on an x-y graph.
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Trajectory of a 2-D Projectile
x
y
Mathematically, the path is defined by a parabola.
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Trajectory of a 2-D Projectile
x
y
For a projectile launched over level ground, the symmetry is apparent.
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Range of a 2-D Projectile
x
y
Range
Definition: The RANGE of the projectile is how far it travels horizontally.
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Maximum height of a projectile
x
y
Range
MaximumHeight
The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.
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Maximum height of a projectile
x
y
Range
MaximumHeight
The vertical velocity component is zero at maximum height.
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Maximum height of a projectile
x
y
Range
MaximumHeight
For a projectile launched over level ground, the maximum height occurs halfway through the flight of the projectile.
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Acceleration of a projectile
g
g
g
g
g
x
y
Acceleration points down at 9.8 m/s2 for the entire trajectory of all projectiles.
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Velocity of a projectile
vo
vf
v
v
v
x
y
Velocity is tangent to the path for the entire trajectory.
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Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
The velocity can be resolved into components all along its path.
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Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
Notice how the vertical velocity changes while the horizontal velocity remains constant.
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Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
Maximum speed is attained at the beginning, and again at the end, of the trajectory if the projectile is launched over level ground.
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vo -
vo
Velocity of a projectile
Launch angle is symmetric with landing angle for a projectile launched over level ground.
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to = 0
t
Time of flight for a projectile
The projectile spends half its time traveling upward…
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Time of flight for a projectile
to = 0
t
2t
… and the other half traveling down.
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Position graphs for 2-D projectiles
x
y
t
y
t
x
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Velocity graphs for 2-D projectiles
t
Vy
t
Vx
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Acceleration graphs for 2-D projectiles
t
ay
t
ax
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Projectile Lab
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Projectile LabThe 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:
1. a table of data 2. a graph of the trajectory3. a calculation of the launch velocity of the
ball obtained from the data
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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.
b) 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?