Motion in Two Dimensions

We begin by studying in greater detail the vector nature of position, velocity, and acceleration. 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

different positions, velocities, and accelerations for a given particle.

4.1 The Position, Velocity, and

Acceleration Vectors

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

We define 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—the average

acceleration of a 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.

4.2 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 generally 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 us that the position vector 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)      

4.3 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

4.4 Uniform Circular Motion

          A car moving in a circular path with constant speed v. Such motion is

called uniform circular motion, and occurs in many situations. It is often surprising

to students to find that even though an object moves at a constant speed in a circular

path, it still has an acceleration. 

 

consider the diagram of the position and velocity vectors in Figure 4.17b. In addition, the figure shows the vector representing the change in positon  

The particle follows a circular path, part of which is shown by the dotted curve.

The particle is at at time and its velocity at that time is . It is at at some later time and its velocity at that time is

In order to calculate the acceleration of the particle, let us begin with the defining equation for average acceleration (Eq. 4.4):

In both Figures 4.17b and 4.17c, we can identify triangles that help us analyze the motion. the two triangles

are similar. (Two triangles are similar if the angle between any two sides is the same for both triangles and if the ratio of the lengths of these sides is the same.)

This enables us to write a relationship between the lengths of the sides for the two triangles

This equation can be solved for and the expression so obtained can be substituted into

to give the magnitude of the average acceleration over the time interval for the particle to move

As  and approach each other,  approaches zero, and the ratio approaches

the speed v. In addition, the average acceleration becomes the instantaneous acceleration at point

Hence in the limit the magnitude of the acceleration is

             

Thus, in uniform circular motion the acceleration is directed inward toward the center

of the circle and has magnitude

                                                                  

4.5 Tangential and Radial Acceleration

This vector can be resolved into two components, based on an origin at the center of the dashed circle: a radial component ar along the radius of the model circle, and a tangential component at perpendicular to this radius.

The total acceleration vector a can be written as the vector sum of

the component vectors:

The tangential acceleration component causes the change in the speed of the

particle. This component is parallel to the instantaneous velocity, and is given by

The radial acceleration component arises from the change in direction of the velocity

vector and is given by

4.6 Relative Velocity and Relative Acceleration

We find that observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle.

As an example

Galilean coordinate transformation

                        (4.21)   

Galilean Velocity Transformation

    (4.22)

Although observers in two frames measure different velocities for the particle, they measure the same acceleration when  is constant. We can verify this by taking the time derivative of Equation 4.22:

That is, the acceleration of the particle measured byan observer in one frame of reference is the same as that measured by any other observer moving with constant velocity relative to the first frame.