Projectile Motion

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 ${\vec{V}}_{0}$ but its acceleration is always the free-fall acceleration $\vec{g}$ , 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 ${\vec{V}}_{0}$ at angle $θ_{0}$ . 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

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

The components $v_{0 x}$ and $v_{0 y}$ can then be found if we know the angle θ₀ between ${\vec{V}}_{0}$ and the positive x direction:

$v_{0 x} = v_{0} cos θ_{0} and v_{0 y} sin θ_{0}$ (4-20)

During its two-dimensional motion, the projectile's position vector $\vec{r}$ and velocity vector $\vec{v}$ change continuously, but its acceleration vector $\vec{a}$ 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 - x₀ from an initial position x₀ is given by Eq. 2-15 with a = 0, which we write as

x - x₀ = v_{0_x}t.

Because v_{0_x} = v₀ cos θ₀, this becomes

x - x₀ = (v₀ sin θ₀)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 - y₀ = v_{0_y}t - $\frac{1}{2} g t^{2} = (v_{0} sin θ_{0}) t - \frac{1}{2} g t^{2}$ (4-22)

where the initial vertical velocity component v_{0_y} is replaced with the equivalent v₀sinθ₀. Similarly, Eqs. 2-11 and 2-16 become

v_y = v₀sin θ₀ - gt (4-23)

and ${v_{y}}^{2} = (v_{0} sin θ_{0}) 2 - 2 g (y - y_{0})$ (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 - x₀ = R in Eq. 4-21 and y - y₀ = 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 θ₀ cos θ₀.

Using the identity sin 2θ₀ = 2 sin θ₀ cos θ₀, we obtain

and R = $\frac{{v_{0}}^{y}}{g}$ sin 2θ₀.

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 2θ₀ = 1, which corresponds to 2θ₀ = 90^o or θ₀ = 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.

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