In Chapter 2 the concepts of displacement, velocity, and acceleration are used to describe an object moving in one dimension. There are also situations in which the motion is along a curved path that lies in a plane. Such two-dimensional motion can be described using the same concepts. In Grand Prix racing, for example, the course follows a curved road, and Figure 3.1 shows a race car at two different positions along it. These positions are identified by the vectors and , which are drawn from an arbitrary coordinate origin. The displacement of the car is the vector drawn from the initial position at time to the final position at time t. The magnitude of is the shortest distance between the two positions. In the drawing, the vectors and are drawn tail to head, so it is evident that is the vector sum of and . (See Sections 1.5 and 1.6 for a review of vectors and vector addition.) This means that , or The displacement here is defined as it is in Chapter 2. Now, however, the displacement vector may lie anywhere in a plane, rather than just along a straight line.

The average velocity of the car in Figure 3.1 between the two positions is defined in a manner similar to that in Equation 2.2, as the displacement divided by the elapsed time : Since both sides of Equation 3.1 must agree in direction, the average velocity vector has the same direction as the displacement . The velocity of the car at an instant of time is its instantaneous velocity . The average velocity becomes equal to the instantaneous velocity in the limit that becomes infinitesimally small : Figure 3.2 illustrates that the instantaneous velocity is tangent to the path of the car. The drawing also shows the vector components and of the velocity, which are parallel to the x and y axes, respectively.

The average acceleration is defined just as it is for one-dimensional motion—namely, as the change in velocity, , divided by the elapsed time : The average acceleration has the same direction as the change in velocity . In the limit that the elapsed time becomes infinitesimally small, the average acceleration becomes equal to the instantaneous acceleration : The acceleration has a vector component along the x direction and a vector component along the y direction.

To understand how displacement, velocity, and acceleration are applied to two-dimensional motion, consider a spacecraft equipped with two engines that are mounted perpendicular to each other. These engines produce the only forces that the craft experiences, and the spacecraft is assumed to be at the coordinate origin when , so that . At a later time t, the spacecraft's displacement is . Relative to the x and y axes, the displacement has vector components of and , respectively.

In Figure 3.3 only the engine oriented along the x direction is firing, and the vehicle accelerates along this direction. It is assumed that the velocity in the y direction is zero, and it remains zero, since the y engine is turned off. The motion of the spacecraft along the x direction is described by the five kinematic variables x, , , , and t. Here the symbol “x” reminds us that we are dealing with the x components of the displacement, velocity, and acceleration vectors. (See Sections 1.7 and 1.8 for a review of vector components.) The variables x, , , and are scalar components (or “components,” for short). As Section 1.7 discusses, these components are positive or negative numbers (with units), depending on whether the associated vector components point in the or the direction. If the spacecraft has a constant acceleration along the x direction, the motion is exactly like that described in Chapter 2, and the equations of kinematics can be used. For convenience, these equations are written in the left column of Table 3.1.

Figure 3.4 is analogous to Figure 3.3, except that now only the y engine is firing, and the spacecraft accelerates along the y direction. Such a motion can be described in terms of the kinematic variables y, , , , and t. And if the acceleration along the y direction is constant, these variables are related by the equations of kinematics, as written in the right column of Table 3.1. Like their counterparts in the x direction, the scalar components, y, , , and , may be positive or negative numbers (with units).

If both engines of the spacecraft are firing at the same time, the resulting motion takes place in part along the x axis and in part along the y axis, as Figure 3.5 illustrates. The thrust of each engine gives the vehicle a corresponding acceleration component. The x engine accelerates the craft in the x direction and causes a change in the x component of the velocity. Likewise, the y engine causes a change in the y component of the velocity. In other words, the x and y motions are independent of each other.

The independence of the x and y motions lies at the heart of two-dimensional kinematics. It allows us to treat two-dimensional motion as two distinct one-dimensional motions, one for the x direction and one for the y direction. Everything that we have learned in Chapter 2 about kinematics in one dimension will now be applied separately to each of the two directions. In so doing, we will be able to describe the x and y variables separately and then bring these descriptions together to understand the two-dimensional picture. Examples 1 and 2 take this approach in dealing with a moving spacecraft.

The following Reasoning Strategy gives an overview of how the equations of kinematics are applied to describe motion in two dimensions, as in Examples 1 and 2.

The biggest thrill in baseball is a home run. The motion of the ball on its curving path into the stands is a common type of two-dimensional motion called “projectile motion.” A good description of such motion can often be obtained with the assumption that air resistance is absent.

Using the equations in Table 3.1, we consider the horizontal and vertical parts of the motion separately. In the horizontal or x direction, the moving object (the projectile) does not slow down in the absence of air resistance. Thus, the x component of the velocity remains constant at its initial value or , and the x component of the acceleration is . In the vertical or y direction, however, the projectile experiences the effect of gravity. As a result, the y component of the velocity is not constant and changes. The y component of the acceleration is the downward acceleration due to gravity. If the path or trajectory of the projectile is near the earth's surface, has a magnitude of . In this text, then, the phrase “projectile motion” means that and equals the acceleration due to gravity. Example 3 and other examples in this section illustrate how the equations of kinematics are applied to projectile motion.

The freely falling package in Example 3 picks up vertical speed on the way downward. The horizontal component of the velocity, however, retains its initial value of throughout the entire descent. Since the plane also travels at a constant horizontal velocity of , it remains directly above the falling package. The pilot always sees the package directly beneath the plane, as the dashed vertical lines in Figure 3.7 show. This result is a direct consequence of the fact that the package has no acceleration in the horizontal direction. In reality, air resistance would slow down the package, and it would not remain directly beneath the plane during the descent.

Figure 3.8 illustrates what happens to two packages that are released simultaneously from the same height, in order to emphasize that the vertical and horizontal parts of the motion in Example 3 occur independently. Package A is dropped from a stationary balloon and falls straight downward toward the ground, since it has no horizontal velocity component . Package B, on the other hand, is given an initial velocity component of in the horizontal direction, as in Example 3, and follows the path shown in the figure. Both packages hit the ground at the same time. Not only do the packages in Figure 3.8 reach the ground at the same time, but the y components of their velocities are also equal at all points on the way down. However, package B does hit the ground with a greater speed than does package A. Remember, speed is the magnitude of the velocity vector, and the velocity of B has an x component, whereas the velocity of A does not. The magnitude and direction of the velocity vector for package B at the instant just before the package hits the ground is computed in Example 4.

An important feature of projectile motion is that there is no acceleration in the horizontal, or x, direction. Conceptual Example 5 discusses an interesting implication of this feature.

Often projectiles, such as footballs and baseballs, are sent into the air at an angle with respect to the ground. From a knowledge of the projectile's initial velocity, a wealth of information can be obtained about the motion. For instance, Example 6 demonstrates how to calculate the maximum height reached by the projectile.

It is also possible to find the total time or “hang time” during which the football in Figure 3.10 is in the air. Example 7 shows how to determine this time.

Another important feature of projectile motion is called the “range.” The range, as Figure 3.10 shows, is the horizontal distance traveled between launching and landing, assuming the projectile returns to the same vertical level at which it was fired. Example 8 shows how to obtain the range.

The range in the previous example depends on the angle at which the projectile is fired above the horizontal. When air resistance is absent, the maximum range results when .

The examples considered thus far have used information about the initial location and velocity of a projectile to determine the final location and velocity. Example 9 deals with the opposite situation and illustrates how the final parameters can be used with the equations of kinematics to determine the initial parameters.

In projectile motion, the magnitude of the acceleration due to gravity affects the trajectory in a significant way. For example, a baseball or a golf ball would travel much farther and higher on the moon than on the earth, when launched with the same initial velocity. The reason is that the moon's gravity is only about one-sixth as strong as the earth's.

Section 2.6 points out that certain types of symmetry with respect to time and speed are present for freely falling bodies. These symmetries are also found in projectile motion, since projectiles are falling freely in the vertical direction. In particular, the time required for a projectile to reach its maximum height H is equal to the time spent returning to the ground. In addition, Figure 3.13 shows that the speed v of the object at any height above the ground on the upward part of the trajectory is equal to the speed v at the same height on the downward part. Although the two speeds are the same, the velocities are different, because they point in different directions. Conceptual Example 10 shows how to use this type of symmetry in your reasoning.

In all the examples in this section, the projectiles follow a curved trajectory. In general, if the only acceleration is that due to gravity, the shape of the path can be shown to be a parabola.

To someone hitchhiking along a highway, two cars speeding by in adjacent lanes seem like a blur. But if the cars have the same velocity, each driver sees the other remaining in place, one lane away. The hitchhiker observes a velocity of perhaps , but each driver observes the other's velocity to be zero. Clearly, the velocity of an object is relative to the observer who is making the measurement.

Figure 3.15 illustrates the concept of relative velocity by showing a passenger walking toward the front of a moving train. The people sitting on the train see the passenger walking with a velocity of , where the plus sign denotes a direction to the right. Suppose the train is moving with a velocity of relative to an observer standing on the ground. Then the ground-based observer would see the passenger moving with a velocity of , due in part to the walking motion and in part to the train's motion. As an aid in describing relative velocity, let us define the following symbols: In terms of these symbols, the situation in Figure 3.15 can be summarized as follows: or According to Equation 3.7^*, is the vector sum of and , and this sum is shown in the drawing. Had the passenger been walking toward the rear of the train, rather than toward the front, the velocity relative to the ground-based observer would have been .

Each velocity symbol in Equation 3.7 contains a two-letter subscript. The first letter in the subscript refers to the body that is moving, while the second letter indicates the object relative to which the velocity is measured. For example, and are the velocities of the Train and Passenger measured relative to the Ground. Similarly, is the velocity of the Passenger measured by an observer sitting on the Train.

The ordering of the subscript symbols in Equation 3.7 follows a definite pattern. The first subscript (P) on the left side of the equation is also the first subscript on the right side of the equation. Likewise, the last subscript (G) on the left side is also the last subscript on the right side. The third subscript (T) appears only on the right side of the equation as the two “inner” subscripts. The colored boxes below emphasize the pattern of the symbols in the subscripts: In other situations, the subscripts will not necessarily be P, G, and T, but will be compatible with the names of the objects involved in the motion.

Equation 3.7 has been presented in connection with one-dimensional motion, but the result is also valid for two-dimensional motion. Figure 3.16 depicts a common situation that deals with relative velocity in two dimensions. Part a of the drawing shows a boat being carried downstream by a river; the engine of the boat is turned off. In part b, the engine is turned on, and now the boat moves across the river in a diagonal fashion because of the combined motion produced by the current and the engine. The list below gives the velocities for this type of motion and the objects relative to which they are measured: The velocity of the boat relative to the water is the velocity measured by an observer who, for instance, is floating on an inner tube and drifting downstream with the current. When the engine is turned off, the boat also drifts downstream with the current, and is zero. When the engine is turned on, however, the boat can move relative to the water, and is no longer zero. The velocity of the water relative to the shore is the velocity of the current measured by an observer on the shore. The velocity of the boat relative to the shore is due to the combined motion of the boat relative to the water and the motion of the water relative to the shore. In symbols, The ordering of the subscripts in this equation is identical to that in Equation 3.7, although the letters have been changed to reflect a different physical situation. Example 11 illustrates the concept of relative velocity in two dimensions.

Sometimes, situations arise when two vehicles are in relative motion, and it is useful to know the relative velocity of one with respect to the other. Example 12 considers this type of relative motion.

The physics of raindrops falling on car windows. While driving a car, have you ever noticed that the rear window sometimes remains dry, even though rain is falling? This phenomenon is a consequence of relative velocity, as Figure 3.19 helps to explain. Part a shows a car traveling horizontally with a velocity of and a raindrop falling vertically with a velocity of . Both velocities are measured relative to the ground. To determine whether the raindrop hits the window, however, we need to consider the velocity of the raindrop relative to the car, not to the ground. This velocity is , and we know that Here, we have used the fact that . Part b of the drawing shows the tail-to-head arrangement corresponding to this vector subtraction and indicates that the direction of is given by the angle . In comparison, the rear window is inclined at an angle with respect to the vertical (see the blowup in part a). When is greater than , the raindrop will miss the window. However, is determined by the speed of the raindrop and the speed of the car, according to . At higher car speeds, the angle becomes too large for the drop to hit the window. At a high enough speed, then, the car simply drives out from under each falling drop!

A primary focus of this chapter has been projectile motion. This section presents two additional examples that serve as a review of the basic features of this type of motion. Example 13 deals with the fact that projectile motion consists of a horizontal and a vertical part, the central idea being that these two parts occur independently of one another. Example 14 stresses the fact that the time variable has the same value for both the horizontal and vertical parts of the motion.