Calc 2 and Calc 3 Vector Problem?

Posted on 2015-09-23 06:26:00 in Math by WILLSON

A ball of radius 1 foot is centered at the origin. An ant, resting at the north pole of the ball, decides to visit the south pole. It heads off in the direction of the positive x-axis at the rate of 1 foot per minute, and continues walking due south at this pace until it reaches the south pole (after p minutes).

However, this ball is spinning counterclockwise about the z-axis at a rate of 1 radian/minute.

(a) Let P(t) be the ant's position after t minutes. Give the coordinates of P(t) in spherical coordinates, cartesian coordinates, and cylindrical coordinates.

(b) Let v(t) and a(t) be the velocity and acceleration of the ant at time t. Find a formula for the dot product v(t) * a(t) [this is v(t) dot a(t)]. At what times t (0<=t<=p) are these vectors orthogonal?

(c) If the ball had NOT been spinning, what would the answer to the last question in (b) have been?

(d) Write down a definite integral that computes the length of the ant's journey. You don't need to evaluate this, but instead give a simple reason why this length is a number between p and v2 * p

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