phy2048-ch4 - Chapter 4 2D and 3D Motion I. Definitions II....

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1 Chapter 4 – 2D and 3D Motion I. Definitions II. Projectile motion III. Uniform circular motion IV. Relative motion Position vector: extends from the origin of a coordinate system to the particle. ) 1 . 4 ( ˆ ˆ ˆ k z j y i x r + + = a ) 2 . 4 ( ˆ ) ( ˆ ) ( ˆ ) ( 1 2 1 2 1 2 1 2 k z z j y y i x x r r r - + - + - = - = Δ a a a I. Definitions Average velocity: ) 3 . 4 ( ˆ ˆ ˆ k t z j t y i t x t r v avg Δ Δ + Δ Δ + Δ Δ = Δ Δ = a a Displacement vector: represents a particle’s position change during a certain time interval.
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2 Instantaneous velocity: ) 4 . 4 ( ˆ ˆ ˆ ˆ ˆ ˆ k dt dz j dt dy i dt dx dt r d k v j v i v v z y x + + = = + + = a a -The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position Instantaneous acceleration: Average acceleration: ) 5 . 4 ( 1 2 t v t v v a avg Δ Δ = Δ - = a a a a ) 6 . 4 ( ˆ ˆ ˆ ˆ ˆ ˆ k dt dv j dt dv i dt dv dt v d k a j a i a a z y x z y x + + = = + + = a a II. Projectile motion Motion of a particle launched with initial velocity, v 0 and free fall acceleration g. - Horizontal motion: a x =0 b v x =v 0x = constant - Vertical motion: a y = -g = constant Range (R): horizontal distance traveled by a projectile before returning to launch height. ) 7 . 4 ( ) cos ( 0 0 0 0 t v t v x x x θ = = - ) 8 . 4 ( 2 1 ) sin ( 2 1 2 0 0 2 0 0 gt t v gt t v y y y - = - = - ) 9 . 4 ( sin 0 0 gt v v y - = ) 10 . 4 ( ) ( 2 ) sin ( 0 2 0 0 2 y y g v v y - - = The horizontal and vertical motions are independent from each other.
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3 - Trajectory: projectile’s path. ) 11 . 4 ( ) cos ( 2 ) (tan cos 2 1 cos sin cos ) 8 . 4 ( ) 7 . 4 ( 2 0 0 2 0 2 0 0 0 0 0 0 0 0 θ v gx x y v x g v x v y v x t - = - = = + 0 0 0 = = y x - Horizontal range: R = x-x 0 ; y-y 0 =0. ) 12 . 4 ( 2 sin cos sin 2 cos 2 1 tan cos 2 1 cos ) sin ( 2 1 ) sin ( 0 cos ) cos ( 0 2 0 2 0 0 0 0 2 2 0 2 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 g v v g R v R g R v R g v R v gt t v v R t t v R = = - = - = - = = = (Maximum for a launch angle of 45º ) Overall assumption: the air through which the projectile moves has no effect on its motion b friction neglected. 122: A third baseman wishes to throw to first base, 127 feet distant. His best throwing speed is 85 mi/h. (a) If he throws the ball horizontally 3 ft above the ground, how far from first base will it hit the ground? (b) From the same initial height, at what upward angle must the third baseman throw the ball if the first baseman is to
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This note was uploaded on 07/30/2011 for the course PHY 2049 taught by Professor Saha during the Spring '08 term at University of Central Florida.

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phy2048-ch4 - Chapter 4 2D and 3D Motion I. Definitions II....

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