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# pg 45 - Quantities characterizing motion in two and three...

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Unformatted text preview: Quantities characterizing motion in two and three dimensions have both magnitude and direction and are desaibed by vectors. Posi— tion, velocity, and acceleration are all vector quantities, related as they are in one dimension: These vector quantities need not have the same direction. In particular, ac— celeration that’s perpendicular to ve- locity changes the direction but not the magnitude of the velocity. Accel- eration that’s coljnear changes only the magnitude of the velocity. In gen- eral, both change. Vectors can be characterized by magnitude and direction or by components. In two dimensions these representations are related by A A: VA3+A; and (belief I AX = A6056 and A), = AsinG Components of motion in two perpendicular directions are inde- pendent. This reduces problems in two and three dimensions to sets of one-dimensional problems that can be solved vvith the methods of Chapter 2. A compact way to express vectors involves unit vectors that have magnitude 1, no units, and point along the coordinate axes: When acceleration is constant, motion is described by vector equations that generalize the one~dimensional equations of Chapter 2: i7=ﬁu+at 1?: An important case of constant—acceleration motion in two dimensions is projectile motion under the inﬂuence of gravity. 5’ = 1: tan 6 — ‘x2 y 0 21102003260 FD + E": + 5:2 Velocity is the rate of change of the position vector 3‘: g dr Acceleration is the rate of change of velocity: 92 dr _. V 5: In uniform circular motion the magnitudes of velocity and acceleration remain constant, but their directions continually change. For an ob« ject moving in a circular path of radius r, the magnitudes of ii and 17 are related by a: vzlr. 17 ...
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