Posted by Lance on Sunday, November 19, 2006 at 2:17pm.
Below is the original question and emailed response from my friend, Dr Ford, a theoretical physicist at Tufts University in Boston.
Question: We have a normal watch.
The big hand is 4cm long and the little hand is 3cm long.
What's the distance between the tips of the hands at the moment they are moving the fastest towards each other.
This is a problem that seems to require calculus. Let A be the angle
between the two hands at some moment. You can write the distance between
the two ends, s, using the law of cosines. Next take the time derivative
of s. The time derivative of A is a constant, as it is the difference in
the angular velocities of the two hands. We now want to find the value
of A where ds/dt is maximum. To find this, we take another derivative
with respect to A and set the result equal to zero. I found that this
happens when cos(A) = sqrt(12/13), or A = 16.1 deg. Putting this back
into the experession for s gave me s = 1.39cm as the distance when the
speed is greatest. If you want more details of the calculation, it would
be easiest for me to mail or Fax you my notes.
hmmm. Will look at that when time is available. Approach sounds reasonable. My brain is not working now, overloaded with birthday parties today.
Something about taking the derivative of s wrt time to get relative velocity. Relative velocity. s is changing direction, and the time derivative of s will yield s in the direction of s...To me, the velocity of s is almost perpendicular to the direction of s, as the velocity of
the hands is perpendicular to the direction of the hands. I will have to think about it. Past history with this problem many years ago is raising a flag.
I can get his notes and email them to you. He's the smartest guy I ever have known. Graduated from Michigan State University with 4.0 in physics then had his doctors in 3 more years. He scored in the top 5 students in the nation on tests for scholaship grants out of high school. Frankly it's over my head. Simple highschool algebra is about my limit. Although 45 years ago I was pretty sharp on Trig and College Algebra. I do enjoy my time on this forum though.
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