**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**

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