Monday, June 24, 2013
SOCS-P (Arons 2.1-2.7)
So, reading Arons and his discussion of a "clock reading" as having zero length in time causes difficulty for me. I immediately struck up a text message stream with my physics professor from college discussing this idea. I will share it tomorrow in class with anyone if you are interested. For me, if I am interested in describing instantaneous velocity and acceleration, I must inherently be discussing TWO TIMES. Therefore an "instant" is by default a very small change in time. Arons expressly states that an "instant" has no "length in time" just as a point has no length in space. I guess this is semantics, but I really think it would hinder a discussion of any kinematic topic other than position. To describe a velocity or an acceleration, two moments in time are needed, and both are the rate of change of one quantity per unit change of another. If I go about describing an instant as being a single time with no "change" than I feel like I would have difficulty describing to a student why the "instantaneous acceleration" of an object at the top of its path is not zero. I always talk about the fact that we are interested in two times, however close together they are, and an object will only be at rest one of those moments. The acceleration will not be zero because the velocity is changing between those TWO times. I have always described it this way, and I talk to my students about the fact that "instantaneous" really is an "average" over a very small time period. I look forward to discussing my difficulties with my group tomorrow. I have named this a "Philosophical" SOCS because I feel like this is a philosophical discussion, not a matter of curriculum or instruction. I could imagine it being an instructional SOCS, but I guess I feel it is more about belief than practice.
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So is the ball stationary at the top of its path for a certain duration? How long of a time span are we talking?
ReplyDeleteWell, again, I think this is semantics. Given my argument in the post, I think the ONLY measured quantity that can exist at a single moment in time of ZERO length is position. Therefore, I would argue (though I wouldn't do this in class) that there is not a single moment in time at which the ball is at rest. The measurement of velocity requires a change in position. We stated in class today that it is impossible to change position without time changing as well (the inverse is not true). So, the only way the velocity could be zero at a particular time would be if there were no change in position over the time interval we were looking at. In the case of the ball, it would change position over any time interval we define. Therefore it cannot be at rest. Now, please understand that for practical (operational) purposes, for a very small time interval, the ratio of the change in position to the time interval would approach zero at the top, and the ratio of the change in velocity (during two short intervals for each velocity) during a different short interval, would approach 9.8 m/s^2 on the surface of the Earth toward the Earth.
ReplyDeleteThis is a great conversation! But kind of outside the scope of the class - which thankfully we all understand. How would we deal with this question - if a ball is thrown upward it has + velocity on the way up and - velocity on the way down. Doesn't it at some point have to have 0 velocity; be at rest? I am not sure that you could measure that velocity but can we know it must be true?
ReplyDeleteThis feels to me like the logical conflict between "to walk out the door, you must always walk halfway out the door... so it follows, if you continue walking halfway forever, you will never get out the door" and the fact that we can walk out the door. Our definition of velocity includes two time, inherently and unavoidably, I think. Therefore it cannot exist in one moment in time. For practical purposes though, it certainly does. However, there is a conflict here with reason, which states that if the ball goes from going up to going down it must stop somewhere in the middle. The question to me is, given our definition of velocity, the term makes no sense for a single time. Hence my discussion on Arons' choice to make it SO clear that an instant has no length in time.
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