2-d motion
DESCRIPTION
2-D Motion. Because life is not in 1-D. General. 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. - PowerPoint PPT PresentationTRANSCRIPT
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2-D MotionBecause life is not in 1-D
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General
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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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A roller coaster rolls down a 20o incline with an acceleration of 5.0 m/s2.
How far horizontally has the coaster traveled in 10 seconds?
How far vertically has the coaster traveled in 10 seconds?
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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
.
a. What are the x and y positions at 5.0 s?
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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
.
b. What are the x and y components of velocity at this time?
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Projectile MotionMoves both horizontally and vertically, subject to acceleration by gravity in vertical direction.
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Projectile MotionFired, 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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Horizontal Component of Velocity
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Vertical Component of Velocity Undergoes accelerated motion Accelerated by gravity (9.8 m/s2 down)
Vy = Vo,y - gty = yo + Vo,yt - 1/2gt2
vy2 = Vo,y
2 - 2g(y – yo)
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Horizontal and Vertical
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Horizontal and Vertical
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Zero Launch angle
A zero launch angle implies a perfectly horizontal launch.
vo
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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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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?
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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 velocityUse 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 velocityThen 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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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 far from the kicker does it land?
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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
tTime 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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Snowballs are thrown with a speed of 13m/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.