哈嘍～二維空間的運動來了鴨～

• projectile?motion:objects taking places in two dimentions if there is no wind.

  Galileo was the first to describe projectile motion accurately. He showed that it could be understood by analyzing the horizontal and vertical components of the motion separately. For convenience, we assume that the motion begins at?time?t?=?0?at the origin of an?xy?coordinate system (so?x0?=?y0?=?0).

• Following Galileo’s ideas, we treat the horizontal and vertical components of velocity and acceleration separately, and we can apply the kinematic equations??to the?x?and?y?components of the motion.

• eg.:

  First we examine the vertical (y) component of the motion. At the instant?the ball leaves the table’s top?(t?=?0),?it has only an?x?component of velocity.

  Once the ball leaves the table (at?t?=?0), it experiences a vertically downward?acceleration?g, the acceleration due to gravity. Thus?vy?is initially zero?Avy0?=?0B ，but increases continually in the downward direction (until the ball hits the?ground). Let us take?y?to be positive upward. Then the acceleration due to gravity?is in the?–y?direction, so?ay?=?–g. From Eq. 2–11a (using?y?in place of?x) we?can write?vy?=?vy0?+?ay?t?=?–gt?since we set?vy0?=?0. The vertical displacement?12?is given by Eq. 2–11b written in terms of?y:?y=y0?+vy0?+2ayt.?12?Given?y=0,v?=0,anda?=–g,theny=–gt.

• In the horizontal direction, on the other hand, there is no acceleration (we are?ignoring air resistance). With?ax?=?0, the horizontal component of velocity,?vx?,remains constant, equal to its initial value,?vx0?, and thus has the same magnitude?at each point on the path. The horizontal displacement (with?ax?=?0) is given by?x=v?t+1a?t2?=v?t

• One result of this analysis, which Galileo himself predicted, is that?an object projected horizontally?will reach the ground in the same time as an object dropped?vertically.This is because the vertical motions are the same in both cases, as shown in Fig. 3–18. Figure 3–19 is a multiple-exposure photograph of an experiment that confirms this.

If an object is projected at an upward angle, as in Fig. 3–20, the analysis is similar, except that now there is an initial vertical component of velocity,?vy0?. Because of the downward acceleration of gravity, the upward component of velocity?vy?gradually decreases with time until the object reaches the highest point on its path, at which point?vy?=?0. Subsequently the object moves down- ward (Fig. 3–20) and?vy?increases in the downward direction, as shown (that is, becoming more negative). As before,?vx?remains constant.

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