The problem that follow involve straight-line motion. The time t is in seconds unless othe...

The problem that follow involve straight-line motion. The time t is in seconds unless otherwise stated.

Problem

The cartesian coordinates of a point P in the x-y plane are related to the polar coordinates of the point by the equations x = r cos θ and v = r sin θ.

(a) Show that the unit vectors i and j are related to the unit vectors er and e0 by

i = cos θ er – sin θ eθ

and

j = sin θ er + cos θ eθ.

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(b) Beginning with the expression for the position vector of P in terms of cartesian coordinates, r = xi + vj. derive Eq. (13.52) for the position vector in terms ol polar coordinates.

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(c) By taking the time derivative of the position vector of point P expressed in terms of cartesian coordinates, derive Eq. (13.55) for the velocity in terms of polar coordinates.