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72 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS

Checkpoint 3

At a certain instant, a fly ball has velocity (the xaxis is horizontal, the

yaxis is upward, and is in meters per second). Has the ball passed its highest point?:v

:v25iˆ4.9jˆ

Figure 4-11The projectile ball always

hits the falling can. Each falls a distance h

from where it would be were there no

free-fall acceleration.

M

Can

h

Zero-

gpath

G

The ball and the can fall

the same distance h.

Figure 4-10One ball is released from rest at
the same instant that another ball is shot
horizontally to the right. Their vertical
motions are identical.

Richard Megna/Fundamental Photographs

The Horizontal Motion

Now we are ready to analyze projectile motion, horizontally and vertically.

We start with the horizontal motion. Because there is no accelerationin the hori-

zontal direction, the horizontal component vxof the projectile’s velocity remains

unchanged from its initial value v 0 xthroughout the motion, as demonstrated in

Fig. 4-12. At any time t, the projectile’s horizontal displacement xx 0 from an

initial position x 0 is given by Eq. 2-15 with a0, which we write as

xx 0 v 0 xt.

Becausev 0 xv 0 cosu 0 , this becomes

xx 0 (v 0 cosu 0 )t. (4-21)

The Vertical Motion

The vertical motion is the motion we discussed in Module 2-5 for a particle in

free fall. Most important is that the acceleration is constant. Thus, the equations

of Table 2-1 apply, provided we substitute gforaand switch to ynotation. Then,

for example, Eq. 2-15 becomes

(4-22)

where the initial vertical velocity component v 0 yis replaced with the equivalent

v 0 sinu 0. Similarly, Eqs. 2-11 and 2-16 become

vyv 0 sinu 0 gt (4-23)

and v^2 y(v 0 sin  0 )^2  2 g(yy 0 ). (4-24)

(v 0 sin  0 )t^12 gt^2 ,

yy 0 v 0 yt^12 gt^2

This feature allows us to break up a problem involving two-dimensional motion

into two separate and easier one-dimensional problems, one for the horizontal

motion (with zero acceleration) and one for the vertical motion (with constant

downward acceleration). Here are two experiments that show that the horizontal

motion and the vertical motion are independent.

Two Golf Balls

Figure 4-10 is a stroboscopic photograph of two golf balls, one simply released and

the other shot horizontally by a spring. The golf balls have the same vertical motion,

both falling through the same vertical distance in the same interval of time.The fact

that one ball is moving horizontally while it is falling has no effect on its vertical mo-

tion;that is, the horizontal and vertical motions are independent of each other.

A Great Student Rouser

In Fig. 4-11, a blowgun G using a ball as a projectile is aimed directly at a can sus-

pended from a magnet M. Just as the ball leaves the blowgun, the can is released. If g

(the magnitude of the free-fall acceleration) were zero, the ball would follow the

straight-line path shown in Fig. 4-11 and the can would float in place after the

magnet released it. The ball would certainly hit the can. However,gisnotzero,

but the ball stillhits the can! As Fig. 4-11 shows, during the time of flight of the

ball, both ball and can fall the same distance hfrom their zero-glocations. The

harder the demonstrator blows, the greater is the ball’s initial speed, the shorter

the flight time, and the smaller the value of h.