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Introduction to 2-Dimensional Motion

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2-Dimensional Motion Definition: motion that occurs with both x and y components. 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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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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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: d = v i t

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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 i,y - at Δy = v i,y t - 1 / 2 at 2 v y 2 = v i,y 2 – 2a(Δy)

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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 -90 o (throwing something straight down) to +90 o (throwing something straight up) and everything in between.

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

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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?

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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?

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General Launch Angle Projectiles

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General launch angle vivi 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 vivi 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 ViVi V i,y = V i sin V i,x = V i cos

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

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Sample problem 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 long was it in the air?

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Sample problem A snowball is thrown with a speed of 13m/s from a roof 7.0 m above the ground. It is thrown in a direction 25 o above the horizontal. a)What is the maximum height obtained by the snowball? b)How far from the base of the cliff does the snowball land?

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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 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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vivi -- vivi Velocity of a projectile Launch angle is symmetric with landing angle for a projectile launched over level ground.

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t i = 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 i = 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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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 35 o above the horizontal. a)How far from the golfer does the ball land?

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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 35 o above the horizontal. The next golfer hits a ball with the same initial speed, but at a greater angle than 45 o. The ball travels the same horizontal distance. What was the initial direction of motion?

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