# 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