This similarity implies that the vertical motion is independent of whether or not 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. This shows that the vertical and horizontal motions are independent. Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls. The ball on the right has an initial horizontal velocity, while the ball on the left has no horizontal velocity. Arrows represent horizontal and vertical velocities at each position. Each subsequent position is an equal time interval. Similarly, how far they walk north is only affected by their motion northward.įigure 3.6 This shows the motions of two identical balls-one falls from rest, the other has an initial horizontal velocity. How far they walk east is only affected by their motion eastward. ![]() The person taking the path shown in Figure 3.5 walks east and then north (two perpendicular directions). We will develop techniques for adding vectors having any direction, not just those perpendicular to one another, in Vector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods.) The Independence of Perpendicular Motions (Note that we cannot use the Pythagorean theorem to add vectors that are not perpendicular. This means that we can use the Pythagorean theorem to calculate the magnitude of the total displacement. Note that in this example, the vectors that we are adding are perpendicular to each other and thus form a right triangle. The third vector is the straight-line path between the two points. These vectors are added to give the third vector, with a 10.3-block total displacement. The second represents a 5-block displacement north. The first represents a 9-block displacement east. For example, observe the three vectors in Figure 3.5. The horizontal and vertical components of the motion add together to give the straight-line path. For two-dimensional motion, the path of an object can be represented with three vectors: one vector shows the straight-line path between the initial and final points of the motion, one vector shows the horizontal component of the motion, and one vector shows the vertical component of the motion. The arrow points in the same direction as the vector. ![]() ![]() The arrow’s length is indicated by hash marks in Figure 3.3 and Figure 3.5. The length of the arrow is proportional to the vector’s magnitude. (Recall that vectors are quantities that have both magnitude and direction.)Īs for one-dimensional kinematics, we use arrows to represent vectors. The fact that the straight-line distance (10.3 blocks) in Figure 3.5 is less than the total distance walked (14 blocks) is one example of a general characteristic of vectors. Figure 3.5 The straight-line path followed by a helicopter between the two points is shorter than the 14 blocks walked by the pedestrian.
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