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Kinematics of a Particle in a Plane Click here to skip to the Polar Description

In a Nut Shell: There are three common descriptions used to describe the motion of a particle. They include the rectangular description, the polar description, and the intrinsic description. Selection of the appropriate description depends on the application.

The Rectangular Description of Motion in the x-y plane - Kinematics

The simplest starting point is to examine the motion of a particle (point mass) in the xy-plane with respect to a fixed frame of reference with origin O. Let r be the position vector to this particle at any time t.

r = x i + y j = position vector of particle at point P at any time t

here i is the unit vector in the x-direction and j is the unit vector in the y-direction

Key Concepts: The derivative of the position vector with respect to time gives the velocity of the particle as measured in the frame of reference. The velocity of the particle is always tangent to its path.

The derivative of the velocity vector with respect to time gives the acceleration of the particle as

measured in the frame of reference. For motion in a plane the acceleration has two components -

one tangent to the path and one normal to the path.

v = dr /dt = dx/dt i + dy/dt j = velocity vector of particle at point P at any time t

The magnitude of the velocity, | v |, is called speed. It is a scalar.

a = dv /dt = d²x/dt² i + d²y/dt² j = acceleration vector of particle at point P at any time t

For rectilinear motion (path is a straight line), the acceleration might be given in terms of

1. time, t 2. speed, v 3. position, x

For a given the acceleration, integration gives speed and position, say along x-axis.

NOTE: By the chain rule a = dv/dt = (dv/dx) (dx/dt) = v dv/dx

Click here for a refresher on dot and vector-products.

Click here for examples. Click here to continue with the intrinsic description.

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