Category: Motion in Two and Three Dimensions

Now we turn from relative motion in one dimension to relative motion in two (and, by extension, three) dimensions. In Fig. 4-21, our two observers are again watching a moving particle P from the origins of reference frames A and B, while B moves at a constant velocity  relative to A. (The corresponding axes of these two frames remain parallel.) Figure 4-21 shows a certain instant…

Suppose you see a duck flying north at 30 km/h. To another duck flying alongside, the first duck seems to be stationary. In other words, the velocity of a particle depends on the reference frame of whoever is observing or measuring the velocity. For our purposes, a reference frame is the physical object to which we attach…

A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the particle is accelerating. That fact may be surprising because we often think of acceleration (a change in velocity) as an increase or decrease in speed. Remember, however, that velocity is a…

Now we are ready to analyze projectile motion, horizontally and vertically. The Horizontal Motion Because there is no acceleration in the horizontal direction, the horizontal component vx of the projectile’s velocity remains unchanged from its initial value v0x throughout the motion, as demonstrated in Fig. 4-13. 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…

We next consider a special case of two-dimensional motion: A particle moves in a vertical plane with some initial velocity 0 but its acceleration is always the free-fall acceleration , 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.…

When a particle’s velocity changes from 1 to 2 in a time interval Δt, its average acceleration  during Δt is If we shrink Δt to zero about some instant, then in the limit  approaches the instantaneous acceleration (or acceleration)  at that instant; that is, If the velocity changes in either magnitude or direction (or both), the particle must have an acceleration. We can write Eq. 4-16 in unit-vector form by substituting Eq. 4-11 for  to obtain We…

If a particle moves from one point to another, we might need to know how fast it moves. Just as in Chapter 2, we can define two quantities that deal with “how fast”: average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation. If a particle moves through a displacement Δ in…

One general way of locating a particle (or particle-like object) is with a position vector , which is a vector that extends from a reference point (usually the origin of a coordinate system) to the particle. In the unit-vector notation of Section 3-5,  can be written where x, y, and z are the vector components of  and the coefficients x, y, and z are its scalar components. The…

In this chapter we continue looking at the aspect of physics that analyzes motion, but now the motion can be in two or three dimensions. For example, medical researchers and aeronautical engineers might concentrate on the physics of the two- and three-dimensional turns taken by fighter pilots in dogfights. (Which parameters of the motion will…