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Chapter+4

# Chapter+4 - Chapter 4 Motion in Two Dimensions 4.1 The...

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Unformatted text preview: Chapter 4 Motion in Two Dimensions 4.1 The position, Velocity, and Acceleration Vectors Study ﬁgure 4.1 very, very carefully... (A) The displacement vector A? of an object is deﬁned as the vector whose magnitude is the shortest distance between the initial and ﬁnal positions of the object, and whose direction points from the initial position to the ﬁnal position. See ﬁgure 4.1. (B) The average velocity vector 172m of an object as it moves from some initial position "i"; to some ﬁnal position i“; during the time interval At = tf —- ti is equal to the ratio of A? to At. Note that the displacement vector A? points in the same direction as the average velocity 17m vector! The instantaneous velocity v is deﬁned as the limit of the average velocity 2—: as At approaches zero. That is, 1? equals the derivative of the position vector with respect to time. f; Ar dl‘ V_1i::m__ At—>0 At Cit (C) The average acceleration vector 5 of ave an object Whose velocity changes by A?! during a time interval At is deﬁned as r V 43 a 2 AV f 1 ave mm m It? At Again, note that gave points in the same direction as Av. The instantaneous acceleration a is deﬁned as the limit of the average acceleration g. as At approaches zero. That is, 5 equals the derivative of the velocity vector with respect to time. [St—>0 At dt 4.2 Two-Dimensional Motion with Constant Acceleration The kinematic equations which describe the motion of a particle moving with constant acceleration are: i— an; 1! <1 at + <l H1 H <3 + 931 H. 2 ..,., 2 "’1 "._"’ view—vi +2a (If ri) If one writes the vectors above in terms of their X- and y-components, then using ”=Xi+yj one obtains two sets of kinematic equations: one set describing the motion along the X-axis, and the other set describing the motion along the y- axis. The kinematic equations are: X-motion: __ 1 xfm—Xi — Vat + _axt2 2 Vf m V. + axt X 1X Viiéx = VIZX + 2 aX (Xfmxi) y—motion: 1 2 yf —yi = Viyt + E ayt ny z Viy +ayt V3}, = V; + 2ay(yf--yi) The parameter that connects the motion along the X-axis With the motion along the y-aXis is the time t. Tell the students to please go over example 4.1 on their own... 4.3 Projectile Motion Given that the free—fall acceleration due to gravity is 5:9.8m/szlzgl then ax * 0 9 constant velocity motion in the horizontal direction. ay = i g '9 free-fall motion in the vertical direction. Study ﬁgure 4.7 and 4.11 very carefully H Hence, for projectile motion in 2-D: x—motion: y—motion: 1 2 Xf ' xi 2 Vixt Yr _ Yi = Viyt + E (igﬁ fo m Vix ny : Viy igt 2 _ 2 ny “" Viy + 2 (ig) (Yr _ Yi) Do example 4.5 and problem # 51. 4.4 Uniform Circular Motion This is the case of motion in two dimensions in which a particle moves at constant speed in a circular path. The period T of the motion is deﬁned as the time required by the particle to complete one revolution. For uniform circular motion, T:27“ V Where r = the radius of the circular path v = the speed (constant) 4.5 Tangential and Radial Acceleration A particle accelerates whenever its velocity vector it? changes over time. (i) A change in the magnitude of the velocity vector results in a tangential component of acceleration at . .. Eli! dt (ii) A change in the direction of the velocity at vector results in a centripetal component of acceleration ac . 2 a = — directed toward the center of the circle. Study ﬁgure 4.19 very carefully... and do example 4.9 In terms of the unit vectors i“ and 8 from polar coordinates, one can write the acceleration vector for circular motion more generally as: _, m, _,, d1?“ Vz,‘ azat+aC=—9 —- Wr dt r 4.5 Relative Velocity The general rules for adding relative velocities are: VBE Z: VBA + VAF + VFE and Where (a) Write each velocity with a double subscript in the proper order, meaning “velocity of (ﬁrst subscript) relative to (second subscript)?” (b) When adding relative velocities, the ﬁrst letter of any subscript is to be the same as the last letter of the preceding subscript. (c) The ﬁrst letter of the subscript of the ﬁrst velocity in the sum, and the second letter of 10 the subscript of the last velocity, are the subscripts, in that order, of the relative velocity represented by the sum. (d) The velocity of body A relative to body B, vAB is the negative of the velocity of body B relative to body A, v1, A. That is, m» VAB : ' VBA Do extra example # 5 from chapter 4. ll © 2004 Brooks/Cole, a division of Thomson Learning, inc. Serway and Jewett, Physics for Scientists and Engineers, file Figure 4.7 and Figure 4.9 11 mm 150 100 50 Vi = 50 mIS ...
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