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2-D Motion Because 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 20 o incline with an acceleration of 5.0 m/s 2. 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/s 2. 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/s 2. b. What are the x and y components of velocity at this time?

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Projectile Motion Moves both horizontally and vertically, subject to acceleration by gravity in vertical direction.

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Projectile Motion 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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Horizontal Component of Velocity

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Vertical Component of Velocity Undergoes accelerated motion Accelerated by gravity (9.8 m/s 2 down) V y = V o,y - gt y = y o + V o,y t - 1/2gt 2 v y 2 = V o,y 2 - 2g(y – y o )

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Horizontal and Vertical

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Zero Launch angle A zero launch angle implies a perfectly horizontal launch. vovo

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

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General launch angle vovo 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 V o,y = V o sin V o,x = V o cos

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Resolving the velocity Then proceed to work problems just like you did with the zero launch angle problems. VoVo V o,y = V o sin V o,x = V o cos

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A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25 o 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 Maximum Height The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.

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Maximum height of a projectile x y Range Maximum Height The vertical velocity component is zero at maximum height.

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Maximum height of a projectile x y Range Maximum Height 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/s 2 for the entire trajectory of all projectiles.

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Velocity of a projectile vovo vfvf v v v x y Velocity is tangent to the path for the entire trajectory.

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Velocity of a projectile vyvy vxvx vxvx vyvy vxvx vyvy vxvx x y vxvx vyvy The velocity can be resolved into components all along its path.

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Velocity of a projectile vyvy vxvx vxvx vyvy vxvx vyvy vxvx x y vxvx vyvy Notice how the vertical velocity changes while the horizontal velocity remains constant.

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Velocity of a projectile vyvy vxvx vxvx vyvy vxvx vyvy vxvx x y vxvx vyvy 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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vovo -- vovo Velocity of a projectile Launch angle is symmetric with landing angle for a projectile launched over level ground.

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t o = 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 t o = 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 25 o 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.

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