A clock's minute hand has length 4 and its hour hand length 3.
What is the distance between the tips at the moment when it is increasing most rapidly?
(In reply to answer
by K Sengupta)
Let us respectively denote the hour hand and the minute hand by the lines CP and CQ. Since the minute hand moves faster than the hour hand, we can assume wlog that CP is fixed and CQ is moving at a constant speed. Let t denote the vector which corresponds to the velocity of the point Q under this assumption. Then, the rate of the change of the distance between P and Q is the component of t in the direction of PQ.
Since t is orthogonal to CQ and the magnitude of t is constant, it
follows that this component is maximal whenever angle CQP = 90 degrees.
Thus, PQ = V(CP^2 - CQ^2) = V(4^2 - 3^2) = V7
= 2.6457513 (correct to 7 places of decimals)
Edited on May 9, 2008, 4:11 pm