The discussion revolves around the relationship between instantaneous and average angular velocity, exploring concepts of angular acceleration and how they affect velocity over time.
Some participants have offered clarifications regarding the definitions of instantaneous and average velocities, while others have provided examples to illustrate these concepts. There appears to be an ongoing exploration of the relationship between these types of velocities without a definitive consensus.
Participants are considering the implications of varying angular acceleration and the definitions of instantaneous versus average velocities, as well as the conditions under which they can be compared. There is an emphasis on understanding the graphical representation of these concepts.
elegysix said:Chances are your velocity was changing. The instantaneous = the average only when it is absolutely constant.
The instantaneous is the average over an arbitrarily short time interval. In general, an average velocity can be over a substantial interval. In notation, d is used for an infinitesimal change, ##\Delta## for a general change. So ##\frac{\Delta x}{\Delta t}## is an average velocity over time ##\Delta t##, while ##\frac{dx}{dt}## is the instantaneous velocity.Aristotle said:Can somebody explain to me the reason why? Thanks!
Aristotle said:Say there was an object that was experiencing a varying angular acceleration decreasing over time to zero. Would that mean its velocity continues to increase/change over time?
Ah I see. So because the velocity is changing over time, we get a curved position vs time graph (representing velocity) to show that its increasing. and so if we take the instant angular velocity at a single time it wouldn't equal with the average angular velocity (between two time intervals) because like you said, that velocity changes for every time, correct? Just wanted to make sure I follow what you're telling me.elegysix said:If the acceleration is nonzero, then velocity is changing, regardless of whether the acceleration varies. It is only when acceleration = 0 that velocity is constant.
Thanks you're the best!elegysix said:It sounds like you've got it right.
I'll provide an example for you.
suppose you have velocities v=(1,2,3,4) at times t=(1,2,3,4)
the average velocity over the whole time is 2.5.
The average velocity over the first 3 seconds is 2.
The instantaneous velocities are only defined at a point, so for instance at t=4, the instantaneous velocity is 4.
make sense?
Right, except there is certain to be some instant in the interval at which the instantaneous velocity equals the average over the interval (mean value theorem).Aristotle said:if we take the instant angular velocity at a single time it wouldn't equal with the average angular velocity (between two time intervals)