How Does the Hour Hand's Velocity and Acceleration Change from Noon to 6pm?

[Chestermiller]

I did an update prior to this current post. I got the right magnitude. However, I cannot make sense of the unit vector.

If the book didn't match my result in post #29, then the book must be wrong. Now for the direction of the unit vector. In what direction is the tip of the hour hand moving at noon (i or j)? In what direction is the tip of the hour hand moving at 6 pm (-i or -j)? (j is the up-down direction, and i is the right-left direction).

 

[negation]

If the book didn't match my result in post #29, then the book must be wrong. Now for the direction of the unit vector. In what direction is the tip of the hour hand moving at noon (i or j)? In what direction is the tip of the hour hand moving at 6 pm (-i or -j)? (j is the up-down direction, and i is the right-left direction).

It struck me while I was in the toilet I made a blunder.

At 12 position: (3.5e-6,0)
Velocity is in the +i direction, j is irrelevant here since its magnitude is 0.
At 6 position: (-3.5e-6i,0)
Velocity is in the -i direction.
p(6) - p(12) = (-7e-6,0)/21600

what I got now is -3.24 e-10. This exactly coincides with the book.

Now, it would be helpful if you could shed some light on the conceptual difference between
v = -2.22e-6 in part(a) and v = rω = 3.5e-6
While I under v = rω implies linear velocity, I do not understand why it is not mathematically valid to use v = -2.22e-6 instead of 3.5e-6

 

[Chestermiller]

It struck me while I was in the toilet I made a blunder.

At 12 position: (3.5e-6,0)
Velocity is in the +i direction, j is irrelevant here since its magnitude is 0.
At 6 position: (-3.5e-6i,0)
Velocity is in the -i direction.
p(6) - p(12) = (-7e-6,0)/21600

what I got now is -3.24 e-10. This exactly coincides with the book.

Now, it would be helpful if you could shed some light on the conceptual difference between
v = -2.22e-6 in part(a) and v = rω = 3.5e-6
While I under v = rω implies linear velocity, I do not understand why it is not mathematically valid to use v = -2.22e-6 instead of 3.5e-6

These are very good questions. The velocity is changing with time (its direction is changing), so the average velocity between the initial time and the final time is only an approximation to the instantaneous velocity at some point between the initial time and the final time. The time at which the average velocity best approximates the instantaneous velocity is at the half-way point, 3 pm. At 3 pm, the instantaneous velocity is -3.5E-6j, while the average velocity is -2.22E-6 j. So the average velocity gives the correct direction for the instantaneous velocity at the half way point, and it is only about 30% lower in magnitude. This is pretty good, considering the huge time interval and the huge change in the displacement vectors over this time interval.

The same approach applies to the acceleration. The average acceleration is -3.23E-11i over the 6 hour interval between noon and 6 pm, while the instantaneous acceleration a at 3pm is -5.08E-11i. So the average acceleration gets the direction of the instantaneous acceleration at the half-way point correct, and it's only about 40% lower in magnitude. This is again pretty good, considering the huge time interval and the huge change in the instantaneous velocity vectors over this time interval.

What we have been dealing with here are finite difference approximation formulas to the instantaneous velocity and instantaneous acceleration from finite changes in displacement and velocity over a finite time interval. The finite difference formula is second order accurate at the half-way point.