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.
Answer:
Hi Lance,
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.
Best regards,
Larry
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.
It seems that your friend, Dr. Ford, has provided a detailed explanation of how to approach the problem of finding the distance between the tips of the hands of a watch at the moment they are moving the fastest towards each other. Here's a breakdown of the steps involved in solving the problem:
1. Define the angle between the two hands as "A" at a specific moment.
2. Use the law of cosines to write the distance between the two ends of the hands, "s," as a function of the angle A.
3. Take the time derivative of "s" to find the rate of change of "s" with respect to time.
4. The time derivative of "A" is a constant, as it represents the difference in the angular velocities of the two hands.
5. Determine the value of "A" where ds/dt (rate of change of "s" with respect to time) is maximum.
6. To find this value of "A," take another derivative of "s" with respect to "A" and set the result equal to zero.
7. Solve for "A" to find the angle at which the speed is the greatest.
8. Substitute this value of "A" back into the expression for "s" to calculate the distance.
Dr. Ford has found that the maximum speed occurs when cos(A) is equal to the square root of 12/13, which corresponds to an angle of approximately 16.1 degrees. Plugging this value back into the expression for "s," Dr. Ford obtained a distance of 1.39 cm.
If you find the calculations too complex or beyond your current understanding, Dr. Ford has offered to provide his notes for a more detailed explanation. If you're interested, you can ask your friend to email or fax those notes to you.
It's great that you appreciate Dr. Ford's expertise and accomplishments in the field of physics. While the problem may seem complex, it can still be interesting to learn about the thought processes and steps involved in solving it. And don't worry if the math is beyond your current abilities, as there are many concepts and areas of expertise within physics that require different levels of understanding. Enjoy your time on the forum and continue to engage with the topics that interest you!