CHAPTER 4

Motion in Two Dimensions

In this chapter we explore the kinematics of a particle moving in two dimensions. We begin by studying in a greater detail the vector nature of position, velocity and acceleration.         

The Position, Velocity, and Acceleration Vectors

                                    

The Position/ Displacement vectors

4.1

displacement is a vector, and the displacement of the particle is the difference between its final position and its initial position. We now define the displacement vector  for the particle of Figure 4.1 as being the difference between its final position vector and its initial position vector:

                                                                             

                  

The direction of  is indicated in Figure 4.1. As we see from the figure, the magnitude of

is less than the distance traveled along the curved path followed by the particle.

Velocity vectors

the average velocity of a particle during the time interval !t as the displacement of the particle divided by the time interval:

                                               

                                                                                                                                  

The  instantaneous velocity v is defined as the limit of the average velocity /        as       approaches zero:

                                   

                               

Acceleration vectors

Knowing the velocity at these points allows us to determine the average acceleration of the particle as it moves is defined as the change in the instantaneous velocity vector    v divided by the time interval   t during which that change occurs:

When the average acceleration of a particle changes during different time intervals,  it is useful to define its instantaneous acceleration. The instantaneous acceleration a is defined as the limiting value of the ratio approaches zero:

                                                 

In other words, the instantaneous acceleration equals the derivative of the velocity vector with respect to time.

Two Dimensional Motion with Constant  Acceleration

Two dimensional motion during which  the acceleration remains constant in both magnitude and direction. It will be useful for analyzing some common types of motion.

The position vector for a particle moving in the xy plane :

                                                              r = x iˆ + y jˆ                                                            (4.6)

·         Where  x, y and r change with time as a particle moves       

·          iˆ and jˆ remain constant                                                                               

velocity vector as a function of  time 

If the position vector is known, the velocity of the particle can be obtained from equations 4.3 and 4.6,Which give

                                                                                

                       (4.7)           

Therefore, substituting from equation  2.9 ,  and 

 into equation 4.7

                                             (4.8)

Graphical Representation of Equations 4.8

  is gene rally not along the direction of either Vi or a because the relationship between of these quantities is a vector expression. We may write in component form :

                                                                 (4.8a)

Position Vector as a Function of time           

The x and y coordinate of a particle moving with constant acceleration are

                       

Substituting these expressions into equation 4.6 , r = x iˆ + y jˆ  which give

                  (4.9) 

This equation tells vector us that the position is the vector sum of the original position  

a displacement   arising from the initial velocity of the particle and a displacement     resulting from the constant acceleration of the particle.

                                                         

Graphical Representation of Equations 4.9

                                                                                                                                               

 is generally not along the direction of Vi or a because it is vector expressions. We may write them in component form

                                 (4.9)      

Projectile Motion

Anyone who has observed a baseball in motion has observed projectile motion. The ball moves in a curved path, and its motion is simple to analyze if we make two assumptions: (1) the free-fall acceleration g is constant over the range of motion and is directed downward,1 and (2) the effect of air resistance is negligible.2 With these assumptions, we find that the path of a projectile, which we call its trajectory, is always a parabola.

Therefore, the initial x and y components of velocity are:

Horizontal Range and Maximum Height of a Projectile

               

The distance R is called the horizontal range of the projectile, and the distance h is its maximum height.

 and