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P U Z Z L E R
This airplane is used by NASA for astronaut training. When it ﬂies along a certain curved path, anything inside the plane that is not strapped down begins to ﬂoat. What causes this strange effect?
(NASA)

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For more information on microgravity in general and on this airplane, visit http://microgravity.msfc.nasa.gov/ and http://www.jsc.nasa.gov/coop/ kc135/kc135.html

c h ap t e r

Motion in Two Dimensions

Chapter Outline
4.1 The Displacement, Velocity, and
Acceleration Vectors

4.2 Two-Dimensional Motion with
Constant Acceleration

4.4 Uniform Circular Motion 4.5 Tangential and Radial Acceleration 4.6 Relative Velocity and Relative
Acceleration

4.3 Projectile Motion 76

4.1

The Displacement, Velocity, and Acceleration Vectors

77

I

nthis chapter we deal with the kinematics of a particle moving in two dimensions. Knowing the basics of two-dimensional motion will allow us to examine — in future chapters — a wide variety of motions, ranging from the motion of satellites in orbit to the motion of electrons in a uniform electric ﬁeld. We begin by studying in greater detail the vector nature of displacement, velocity, andacceleration. As in the case of one-dimensional motion, we derive the kinematic equations for two-dimensional motion from the fundamental deﬁnitions of these three quantities. We then treat projectile motion and uniform circular motion as special cases of motion in two dimensions. We also discuss the concept of relative motion, which shows why observers in different frames of reference may measure differentdisplacements, velocities, and accelerations for a given particle.
y

4.1

THE DISPLACEMENT, VELOCITY, AND ACCELERATION VECTORS
O

ti ri rf

∆r

tf

In Chapter 2 we found that the motion of a particle moving along a straight line is completely known if its position is known as a function of time. Now let us extend this idea to motion in the xy plane. We begin by describing theposition of a particle by its position vector r, drawn from the origin of some coordinate system to the particle located in the xy plane, as in Figure 4.1. At time ti the particle is at point , and at some later time tf it is at point . The path from to is not necessarily a straight line. As the particle moves from to in the time interval t t f t i , its position vector changes from ri to rf . As welearned in Chapter 2, displacement is a vector, and the displacement of the particle is the difference between its ﬁnal position and its initial position. We now formally deﬁne the displacement vector r for the particle of Figure 4.1 as being the difference between its ﬁnal position vector and its initial position vector: r rf ri (4.1) The direction of r is indicated in Figure 4.1. As we see fromthe ﬁgure, the magnitude of r is less than the distance traveled along the curved path followed by the particle. As we saw in Chapter 2, it is often useful to quantify motion by looking at the ratio of a displacement divided by the time interval during which that displacement occurred. In two-dimensional (or three-dimensional) kinematics, everything is the same as in one-dimensional kinematicsexcept that we must now use vectors rather than plus and minus signs to indicate the direction of motion. We deﬁne the average velocity of a particle during the time interval t as the displacement of the particle divided by that time interval: v r t (4.2)

Path of particle
x

Figure 4.1

A particle moving in the xy plane is located with the position vector r drawn from the origin to theparticle. The displacement of the particle as it moves from to in the time interval t t f ti is equal to the vector r rf ri .
Displacement vector

Average velocity

Multiplying or dividing a vector quantity by a scalar quantity changes only the magnitude of the vector, not its direction. Because displacement is a vector quantity and the time interval is a scalar quantity, we conclude that the...