Motion in a plane

1. Introduction

  • In previous chapter we have learned about the motion of any particle along a straight line
  • Straight line motion or rectilinear motion is motion in one dimension.Now in this chapter ,we will consider both motion in two dimension and three dimension.
  • In two dimensional motion path of the particle is constrained to lie in a fixed plane.Example of such motion motion are projectile shot from a gun ,motion of moon around the earth,circular motion and many more.
  • To solve problems of motion in a plane,we need to generalize kinematic language of previous chapter to a more general using vector notations in two and three dimensions.

2.Average velocity

  • Consider a particle moving along a curved path in x-y plane shown below in the figue
  • Suppose at any time,particle is at the point P and after some time 't' is at point Q where points P and Q represents the position of particle at two different points.

    Displacement vector in xy coordinate system

  • Position of particle at point P is described by the Position vector r from origin O to P given by
    where x and y are components of r along x and y axis
  • As particle moves from P to Q,its displacement would be would be Δr which is equal to the difference in position vectors r and r'.Thus
    Δr = r'-r = (x'i+y'j)-(xi+yj) = (x'-x)i+(y'-y)j = Δxi+Δyj (1)
    where Δx=(x'-x) and Δy=(y'-y)
  • If Δt is the time interval during which the particle moves from point P to Q along the curved path then average velocity(vavg) of particle is the ratio of displacement and corresponding time interval

    average velocity equation
    since vavgr/Δt , the direction of average velocity is same as that of Δr
  • Magnitude of Δr is always the straight line distance from P to Q regardless of any shape of actual path taken by the particle.
  • Hence average velocity of particle from point P to Q in time interval Δt would be same for any path taken by the particle.


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