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 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 4 2 Average velocity r x y z vavg i j k t t t t 4 3 1 Instantaneous velocity dr dx dy dz i v vxi v y j vz k j 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 dv x dv y dv z a a x i a y j a z k i j k 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 2 Trajectory projectile s path 4 7 4 8 t y tan 0 x x0 y0 0 x x x 1 y v 0 sin 0 g v 0 cos 0 v 0 cos 0 2 v 0 cos 0 gx 2 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 0 v0 sin 0 t R 1 2 R 1 R 1 R2 R tan 0 g 2 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 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 3600s 1mi 3 feet 0 305m 0 91m 1 foot v0 h 3ft B1 B3 xmax 0 x xB1 38 7m Horizontal movement Vertical movement xmax x0 v0 xt 1 2 gt 2 0 0 91m 4 9t 2 t 0 43s y y0 v0 y t xmax 0 38t 38m s 0 43s 16 4 m 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 7m 38 7 v0 x v0 cos t 1s t 38 cos 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 v0 h 3ft x B3 38 7m B1 3 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 Range R 1 45 v0 v02 sin 2 0 d max at h 0 g 2 45 45 x x R R R R sin a b sin a cos b cos a sin b sin a b sin a cos b cos a sin b R R v02 v2 sin 2 45 0 sin 90 2 g g v02 g sin 2 45 v02 g sin 90 2 v02 v2 sin 90 cos 2 cos 90 sin 2 0 cos 2 g g v02 2 0 sin 90 cos 2 cos 90 sin 2 vg cos 2 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 Acceleration centripetal a Period of revolution T 2 r v vy vx v2 r 4 13 4 14 v y p v x p i j 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 i j a v y i v x j cos i sin j dt r dt r dt r r r r v2 v2 cos 2 sin 2 r r a y sin a directed along radius tan tan a x cos a a x2 a 2y 4 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 v2 In 3s the velocity is reversed the cat reaches the opposite side of the circle x v1 y v 32 4 2 5m s r 2 r T 3s r 4 77 m v 5m s 2 2 2 v 25m s 5 23m s 2 ac r 4 77m v v 6m s i 8m s j aavg 2 1 2m s 2 i 2 67 m s 2 j t 3s aavg 3 33m s 2 IV 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 aPA 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 5 75 A sled moves in the negative x direction at speed vs while a ball of ice is shot from the sled with a velocity v0 v0xi v0yj relative to the sled When the ball lands its horizontal displacement xbg relative to the ground from its launch position to its landing position is measured The figure gives xbg as a function of vs Assume it lands at approximately its launch height What are the values of a v0x and b v0y The ball …
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