**Figure 4-8** A stroboscopic photograph of a yellow tennis ball bouncing off a hard
surface. Between impacts, the ball has projectile motion.

We next consider a special case of two-dimensional motion: A particle moves in a
vertical plane with some initial velocity
${\overrightarrow{V}}_{0}$
but its acceleration is always the free-fall acceleration
$\overrightarrow{g}$, which is downward. Such a particle
is called a *projectile* (meaning that it is projected or launched), and its motion is
called *projectile motion*. A projectile might be a tennis ball (Fig. 4-8) or baseball in
flight, but it is not a duck in flight. Many sports involve the study of the projectile motion
of a ball. For example, the racquetball player who discovered the Z-shot in the 1970s easily
won his games because of the ball's perplexing flight to the rear of the court.

**Figure 4-9** The *projectile motion* of an object launched into the air at the origin of a
coordinate system and with launch velocity
${\overrightarrow{V}}_{0}$
at angle ${\theta}_{0}$. The motion is a combination of
vertical motion (constant acceleration) and horizontal motion (constant velocity), as shown by
the velocity components

Our goal here is to analyze projectile motion using the tools for two-dimensional motion making the assumption that air has no effect on the projectile. Figure 4-9, which we shall analyze soon, shows the path followed by a projectile when the air has no effect. The projectile is launched with an initial velocity that can be written as

${v}_{0}={v}_{0x}\hat{i}+{v}_{0y}\hat{j}$ (4-19)

The components ${v}_{0x}$
and ${v}_{0y}$
can then be found if we know the angle θ_{0} between
${\overrightarrow{V}}_{0}$
and the positive *x* direction:

${v}_{0x}={v}_{0}cos{\theta}_{0}\text{and}{v}_{0y}sin{\theta}_{0}$ (4-20)

During its two-dimensional motion, the projectile's position vector
$\overrightarrow{r}$
and velocity vector
$\overrightarrow{v}$
change continuously, but its acceleration vector
$\overrightarrow{a}$
is constant and *always* directed vertically downward.
The projectile has *no* horizontal acceleration.

Projectile motion, like that in Figs. 4-8 and 4-9, looks complicated, but we have the following simplifying feature (known from experiment):

In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.

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.

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

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

**Figure 4-11** The projectile ball always hits the falling can. Each falls a distance
*h* from where it would be were there no free-fall acceleration.

**A Great Student Rouser**

In Fig. 4-11, a blowgun G using a ball as a projectile is aimed directly at a can
suspended 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, g is *not zero*,
but the ball *still* hits the can! As Fig. 4-11 shows, during the time of flight
of the ball, both ball and can fall the same distance h from their zero-g locations.
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.

**Figure 4-12** The vertical component of this skateboarder's velocity is changing
but not the horizontal component, which matches the skateboard's velocity. As a result, the
skateboard stays underneath him, allowing him to land on it.

**The Horizontal Motion**

Now we are ready to analyze projectile motion, horizontally and vertically. We
start with the horizontal motion. Because there is no acceleration in the horizontal
direction, the horizontal component v_{x} of the projectile's velocity remains
unchanged from its initial value v_{0x} throughout the motion, as demonstrated
in Fig. 4-12. At any time *t*, the projectile's horizontal displacement x - x_{0}
from an initial position x_{0} is given by Eq. 2-15 with a = 0, which we write as

x - x_{0} = v_{0x}t.

Because v_{0x} = v_{0} cos θ_{0}, this becomes

x - x_{0} = (v_{0} sin θ_{0})t.

**The Vertical Motion**

The vertical motion is 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 -g for a and switch to y notation.Then, for example, Eq. 2-15 becomes

y - y_{0} =
v_{0y}t -
$\frac{1}{2}g{t}^{2}=({v}_{0}sin{\theta}_{0})t-\frac{1}{2}g{t}^{2}$ (4-22)

where the initial vertical velocity component v_{0y} is replaced
with the equivalent v_{0}sinθ_{0}. Similarly,
Eqs. 2-11 and 2-16 become

v_{y} = v_{0}sin θ_{0} - gt (4-23)

and ${{v}_{y}}^{2}=({v}_{0}sin{\theta}_{0})2-2g(y-{y}_{0})$ (4-24)

As is illustrated in Fig. 4-9 and Eq. 4-23, the vertical velocity component behaves
just as for a ball thrown vertically upward. It is directed upward initially, and its
magnitude steadily decreases to zero, *which marks the maximum height of the path*.
The vertical velocity component then reverses direction, and its magnitude becomes
larger with time.

**The Horizontal Range**

The horizontal range R of the projectile is the *horizontal* distance the projectile
has traveled when it returns to its initial height (the height at which it is launched).
To find range R, let us put x - x_{0} = R in Eq. 4-21 and y - y_{0} = 0 in
Eq. 4-22, obtaining

R = (v_{0 cos θ)t}

and 0 = (v_{0 cos θ)t - $\frac{1}{2}g{t}^{2}$.}

Eliminating t between these two equations yields

and R =
$\frac{{{v}_{0}}^{y}}{g}$
sin θ_{0} cos θ_{0}.

Using the identity sin 2θ_{0} = 2 sin θ_{0} cos θ_{0}, we obtain

and R =
$\frac{{{v}_{0}}^{y}}{g}$
sin 2θ_{0}.

This equation does not give the horizontal distance traveled by a projectile when
the final height is not the launch height. Note that R in Eq. 4-26 has its maximum
value when sin 2θ_{0} = 1, which corresponds to 2θ_{0}
= 90^{o} or θ_{0} = 45^{o}.

• The horizontal range R is maximum for a launch angle of 45^{o}.

However, when the launch and landing heights differ, as in many sports, a launch
angle of 45^{o} does not yield the maximum horizontal distance.

**The Effects of the Air**

**Figure 4-13** (I) The path of a fly ball calculated
by taking air resistance into account.
(II) The path the ball would follow in a vacuum, calculated by the methods of this
chapter. See Table 4-1 for corresponding data.

We have assumed that the air through which the projectile moves has no effect
on its motion. However, in many situations, the disagreement between our calculations
and the actual motion of the projectile can be large because the air resists
(opposes) the motion. Figure 4-13, for example, shows two paths for a fly ball that
leaves the bat at an angle of 60^{o} with the horizontal and an initial speed of
44.7 m/s. Path I (the baseball player's fly ball) is a calculated path that
approximates normal conditions of play, in air. Path II (the physics professor's fly
ball) is the path the ball would follow in a vacuum.

Table 4-1 Two Fly Balls

**About the Authors**

David Halliday was an American physicist known for his physics textbooks, *Physics and
Fundamentals of Physics*, which he wrote with Robert Resnick. Both textbooks have
been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

The 10th edition of Halliday's Fundamentals of Physics, Extended building upon previous issues by offering several new features and additions. The new edition offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition.

Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. The Halliday content is widely accepted as clear, correct, and complete. The end-of-chapters problems are without peer. The new design, which was introduced in 9e continues with 10e, making this new edition of Halliday the most accessible and reader-friendly book on the market.

A Reader says,"As many reviewers have noted, this is a great physics book used widely in university technical programs as a first course in technical physics, with calculus. I find it is the one book I start with when trying to understand physical concepts at a useful but basic level. It has broad coverage and is well written . To go beyond this book requires specialized books on each topic of interest (electromagnetics, quantum mechanics, thermodynamics, etc.)."

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