Chapter 4 2D and 3D Motion I Definitions II Projectile motion III Uniform circular motion IV Non uniform circular motion V Relative motion I Definitions Position vector extends from the origin of a coordinate system to the particle r xi y j zk 4 1 Displacement vector represents a particle s position change during a certain time interval r r2 r1 x2 x1 i y2 y1 j z 2 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 i j k v vxi v y j vz k dt dt dt dt 4 4 The direction of the instantaneous velocity of a particle is always tangent to the particle s path at the particle s 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 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 constant 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 constant 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 projectile s path x0 y0 0 1 x x x 4 7 4 8 t g y v 0 sin 0 2 v 0 cos 0 v 0 cos 0 v 0 cos 0 gx 2 y tan 0 x 2 v 0 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 1 1 R R R2 R tan 0 g 2 0 v0 sin 0 t gt v0 sin 0 g 2 v0 cos 2 0 2 v0 cos 0 2 v0 cos 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 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 y mi 1h 1609m 85 38m s h 3600 s 1mi 3 feet 0 305m 0 91m 1 foot v 0 h 3ft B1 B3 xmax 0 xB1 38 7m x Horizontal movement Vertical movement xmax x0 v0 x t 1 y y0 v0 y t gt 2 2 0 0 91m 4 9t 2 t 0 43s xmax 0 38t 38m s 0 43s 16 4m from B3 The ball will hit ground at 22 3 m from B1 1 2 38 sin gt v0 y 4 9t v0 sin t 2 4 9 38 7 m 38 7 v0 cos t 1s v0 x 38 cos t 38 7 38 sin 189 63 1444 sin cos 38 cos 4 9 0 13 0 5 sin 2 7 6 y y0 0 v0 y t y v 0 h 3ft B3 38 7m B1 x N7 In Galileo s 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 v v02 Range R sin 2 0 d max at h 0 g 2 45 0 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 Velocity tangent to circle in the direction of motion v2 a r Acceleration centripetal Period of revolution T 2 r v vy vx 4 13 4 14 v y p v x p j i v v x 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 i cos 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 54 A cat rides a merry go round while turning with uniform circular motion At time t1 2s the cat s velocity is v1 3m s i 4m s j measured on an horizontal xy coordinate system At time t2 5s its velocity is v2 3m s i 4m s j What are a the magnitude of the cat s centripetal acceleration and b the cat s average acceleration during the time interval t2 t1 v 2 x In 3s the velocity is reversed the cat reaches the opposite side of the circle v1 y v 32 4 2 5m s r 2 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 2m s 2 i 2 67 m s 2 j aavg t 3s aavg 3 33m s 2 IV Non Uniform circular motion A particle moves with varying speed in a circular path The acceleration has two components Radial ar ac v2 r Tangential at dv dt at causes the change in the speed of the particle a ar2 at2 d v v2 a at ar r dt r In uniform circular motion v constant at 0 a ar V Relative motion Particle s 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 75 A sled moves in the negative x direction at speed vs while a ball of ice …
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