Tuesday, July 17, 2007

Irodov Problem 1.26

The velocity vector of the point is given by the time derivative of its position vector r as,

a) The distance traversed by the point is the integral of the speed or magnitude of the velocity vector and thus is given by,

Bascially v is a vector that is rotating at a constant angular speed of w in a circle. At time t=0, it is oriented along the x-axis and then it rotates counter-clockwise with time (this can be seen simply by seeing that after 90 degree rotation v will oriented towards the +ive y axis). The acceleration vector is also a vector thats rotating at a constant angular speed of w. At time t=0 this one however is oriented along the y-axis and then it rotates counter-clockwise. In other words the acceleration vector constantly leads the velocity vector by 90 degrees.

This can also be mathematically derived by determining the angle between the two based on the dot product as,

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