# Chapter 4 - 2D and 3D Motion - Valencia

Physics 2048 Spring 2008 Lecture #4 Chapter 4 motion in 2D and 3D Chapter 4 2D and 3D Motion I. Definitions II. Projectile motion III. Uniform circular motion IV. Relative motion I. Definitions Position vector: extends from the origin of a coordinate system to the particle. r xi yj zk (4.1) Displacement vector: represents a particles position change during a certain time interval. r r2 r1 ( x2 x1 )i ( y2 y1 ) j ( z2 z1 )k Average velocity: r x y z vavg i j k t t t t (4.3) (4.2) Instantaneous velocity:

d r dx dy dz v v x i v y j v z k i j k dt dt dt dt (4.4) -The direction of the instantaneous velocity of a particle is always tangent to the particles path at the particles position v2 v1 v t t (4.5) dv y dv d v j dv z k a a x i a y j a z k x i dt dt dt dt (4.6)

Average acceleration: aavg Instantaneous acceleration: acceleration II. Projectile motion Motion of a particle launched with initial velocity, v0 and free fall acceleration g. The horizontal and vertical motions are independent from each other. - Horizontal motion: ax=0 vx=v0x= cte x x0 v0 x t (v0 cos 0 )t (4.7) Range (R): horizontal distance traveled by a projectile before returning to launch height. - Vertical motion: ay= -g y y0 v0 y t 1 2 1 gt (v0 sin 0 )t gt 2 2 2 v y v0 sin 0 gt (4.9) (4.8) v y 2 (v0 sin 0 ) 2 2 g ( y y0 ) (4.10) - Trajectory: projectiles path. x0 y0 0 x x

1 x ( 4.7) ( 4.8) t y v0 sin 0 g v0 cos 0 v0 cos 0 2 v0 cos 0 gx 2 y (tan 0 ) x 2(v0 cos 0 ) 2 2 ( 4.11) - Horizontal range: R = x-x0; y-y0=0. R (v0 cos 0 )t t R v0 cos 0 2 1 2 R 1 R 1 R2 R tan 0 g 2 0 (v0 sin 0 )t gt (v0 sin 0 ) g 2 v0 cos 0 2 v0 cos 0 2 v0 cos 2 0 2 sin 0 cos 0 2 v02

R v0 sin 2 0 g g (4.12) (Maximum for a launch angle of 45 ) Overall assumption: the air through which the projectile moves has no effect on its motion friction neglected. ## In Galileos Two New Sciences, the author states that for elevations (angles of projection) which exceed or fall short of 45 by equal amounts, the ranges are equal Prove this statement. y 45 1 45 v0 v02 Range : R sin 2 0 d max at h 0 g 2 45 =45 x=R=R? x v02 v02 R sin 2 45 sin 90 2 g g v02 v02 R ' sin 2 45 sin 90 2

g g sin(a b) sin a cos b cos a sin b sin(a b) sin a cos b cos a sin b v02 v02 R sin 90 cos(2 ) cos 90 sin( 2 ) cos(2 ) g g v02 v02 R ' sin 90 cos(2 ) cos 90 sin( 2 ) cos( 2 ) g g

III. Uniform circular motion Motion around a circle at constant speed. Magnitude of velocity and acceleration constant. Direction varies continuously. continuously -Velocity: tangent to circle in the direction of motion. v2 a r - Acceleration: centripetal - Period of revolution: revolution T 2r v v0y v0x (4.13) (4.14) v y p v x p i j v vx i v y j ( v sin )i (v cos ) j r r v2 dv v dy p v dx p v v v 2 j

a i j v i v j cos i sin y x dt r dt r dt r r r r v2 v2 2

2 a cos sin r r a y sin a directed along radius tan tan a x cos a x2 a 2y 1- A cat rides a merry-go-round while turning with uniform circular motion. At time t 1= 2s, the cats velocity is: v1= (3m/s)i+(4m/s)j, measured on an horizontal xy coordinate system. At time t 2=5s its velocity is: v2= (-3m/s)i+(-4m/s)j. What are (a) the magnitude of the cats centripetal acceleration and (b) the cats average acceleration during the time interval t 2-t1? v2 x In 3s the velocity is reversed the cat reaches the opposite side of the circle v1 y v 32 4 2 5m / s 2r r T 3s r 4.77 m v 5m / s v 2 25m 2 / s 2 ac 5.23m / s 2 r 4.77 m

v2 v1 ( 6m / s)i (8m / s ) j aavg ( 2m / s 2 )i (2.67m / s 2 ) j t 3s aavg 3.33m / s 2 IV. Relative motion Particles velocity depends on reference frame vPA vPB vBA (4.15) 1D Frame moves at constant velocity 0 d d d (vPA ) (vPB ) (vBA ) a PA a PB dt dt dt (4.16) Observers on different frames of reference measure the same acceleration for a moving particle if their relative velocity is constant. constant

## Recently Viewed Presentations

• Therefore, if the subject of your poem were of a flower, then the poem would be shaped like a flower. If it were of a fish, then the poem would take on the shape of a fish FREE VERSE Free...
• Give several reasons for starting a new business from scratch rather than buying a franchise or an existing business. Distinguish among the different types and sources of startup ideas. Describe external and internal analyses that might shape new venture opportunities....
• Check with AICT before putting a whole whack of colour transitions on the poster. Adobe Illustrator & Photoshop do not have this limitation. You may use any graphic. design software to create. your poster - but replicate . this template...
• The Human Body Did you Know??!! When you sneeze all body functions stop, even your heart Your brain is 80% water A human has 60,000 miles of blood vessels in their body The lining of your digestive system is shed...
• Rotational displacement of transverse and sagittal axes significantly differed and increased when participants tried to rotate, especially on dominant hip. Decreased axial trunk ROM on dominant hip was related to stiffened passive structures of hip joint.
• PCS. Hawaii. Tampa. Shreveport. Alaska. FXVA. Myrtle Beach. Michigan. Realogy. HHI. SHPS. Door County. Mississippi. Gatlinburg. Hawaii Electric. Tremendous Interest. As a result of our recent work in introducing this in the last 60 days, here's those either committed or...
• Membranes, transport, and signaling Andy Howard Introductory Biochemistry Wednesday 8 October 2014 10/08/2014 Membranes, Transport, Signaling * * * * * * * 10/08/2014 Membranes, Transport, Signaling p. * of 53 Extracellular Signals Internal behavior of cells modulated by external...
• Marketing. Satisfies the needs and wants of consumers through exchange. Based on four "P's" Product. Price (establishing the price of the product) Placement (placing the product in attractive and convenient sales locations) Promotion (advertising) Functions of Business