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Motion Along A Straight Line
Key Ideas ● The position x of a particle on an x axis locates the particle with respect to the origin, or zero point, of the axis. ● The position is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin. The positive direction on an axis is the direction of increasing positive numbers; the opposite direction is the negative direction on the axis. ● The displacement Δx of a particle is the change in its position: Δx = x2 − x1. ● Displacement is a vector quantity. It is positive if the particle has moved in the positive direction of the x axis and negative if the particle has moved in the negative direction. ● When a particle has moved from position x1 to position x2 during a time interval Δt = t2 − t1, its average velocity during that interval is vavg = Δx = x2 − x1 Δt t2 − t1 ● The algebraic sign of vavg indicates the direction of motion (vavg is a vector quantity). Average velocity does not depend on the actual distance a particle moves, but instead depends on its original and final positions. ● On a graph of x versus t, the average velocity for a time interval Δt is the slope of the straight line connecting the points on the curve that represent the two ends of the interval. ● The average speed savg of a particle during a time interval Δt depends on the total distance the particle moves in that time interval: savg =total distance Δt .
Key Ideas ● Scalars, such as temperature, have magnitude only. They are specified by a number with a unit (10°C) and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra. ● Two vectors a → and b → may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum s →. To subtract b → from a →, reverse the direction of b → to get ‒b → ; then add ‒b → to a →. Vector addition is commutative and obeys the associative law. ● The (scalar) components ax and ay of any two-dimensional vector a → along the coordinate axes are found by dropping perpendicular lines from the ends of a → onto the coordinate axes. The components are given by ax = a cos θ and ay = a sin θ, where θ is the angle between the positive direction of the x axis and the direction of a →. The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation of the vector a → with a = √a2x + a2y and tan θ = ay ax .
Motion in Two and Three Dimensions
Key Ideas ● The location of a particle relative to the origin of a coordinate system is given by a position vector r →, which in unit-vector notation is r → = xiˆ + yjˆ + zkˆ . Here xiˆ, yjˆ, and zkˆ are the vector components of position vector r →, and x, y, and z are its scalar components (as well as the coordinates of the particle). ● A position vector is described either by a magnitudeand one or two angles for orientation, or by its vector or scalar components. ● If a particle moves so that its position vector changes from r1 → to r2→ , the particle’s displacement Δr → is Δr → = r2 → − r1 →. The displacement can also be written as Δr → = (x2 − x1)iˆ + (y2 − y1)jˆ + (z2 − z1)kˆ = Δxiˆ + Δyjˆ + Δzkˆ .
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