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phy2048-ch4

# 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 + + = arrowrightnosp ) 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 arrowrightnosp arrowrightnosp arrowrightnosp 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 = arrowrightnosp arrowrightnosp 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 + + = = + + = arrowrightnosp arrowrightnosp -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 - = arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp ) 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 + + = = + + = arrowrightnosp arrowrightnosp II. Projectile motion Motion of a particle launched with initial velocity, v 0 and free fall acceleration g. - Horizontal motion: a x =0 barb2right 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 - = - = - θ ) 9 . 4 ( sin 0 0 gt v v y - = θ ) 10 . 4 ( ) ( 2 ) sin ( 0 2 0 0 2 y y g v v y - - = θ The horizontal and vertical motions are independent from each other.
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 θ θ θ θ 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 θ θ θ θ θ θ θ θ θ θ θ 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 barb2right 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 catch it 3 ft above the ground? (c) What will be the time of flight in that case?

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

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