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This similarity implies vertical motion is independent of whether the ball is moving horizontally. It is remarkable that for each flash of the strobe, the vertical positions of the two balls are the same. Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls, which shows the vertical and horizontal motions are independent. The ball on the right has an initial horizontal velocity whereas the ball on the left has no horizontal velocity. Arrows represent the horizontal and vertical velocities at each position. Each subsequent position is an equal time interval. Similarly, how far she walks north is affected only by her motion northward.įigure 4.8 A diagram of the motions of two identical balls: one falls from rest and the other has an initial horizontal velocity. How far she walks east is affected only by her motion eastward. The woman taking the path from A to B may walk east for so many blocks and then north (two perpendicular directions) for another set of blocks to arrive at B. To illustrate this concept with respect to displacement, consider a woman walking from point A to point B in a city with square blocks. Thus, the motion of an object in two or three dimensions can be divided into separate, independent motions along the perpendicular axes of the coordinate system in which the motion takes place. Motion along the x direction has no part of its motion along the y and z directions, and similarly for the other two coordinate axes. When we look at the three-dimensional equations for position and velocity written in unit vector notation, (Figure) and (Figure), we see the components of these equations are separate and unique functions of time that do not depend on one another. The Independence of Perpendicular Motions We take the radius of Earth as 6370 km, so the length of each position vector is 6770 km. The vector between them is the displacement of the satellite. The position vectors are drawn from the center of Earth, which we take to be the origin of the coordinate system, with the y-axis as north and the x-axis as east. Although satellites are moving in three-dimensional space, they follow trajectories of ellipses, which can be graphed in two dimensions. (Figure) shows the surface of Earth and a circle that represents the orbit of the satellite. We then use unit vectors to solve for the displacement.
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This will aid in our understanding of the displacement. We make a picture of the problem to visualize the solution graphically. What is the magnitude and direction of the displacement vector from when it is directly over the North Pole to when it is at A satellite is in a circular polar orbit around Earth at an altitude of 400 km-meaning, it passes directly overhead at the North and South Poles.