# Why does the Average angular velocity and Instantaneous angular velocity give different results?

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1. Nov 3, 2014

### Aristotle

• Note that use of the homework template is mandatory in the homework forums.
Can somebody explain to me the reason why? Thanks!

2. Nov 3, 2014

### elegysix

Chances are your velocity was changing. The instantaneous = the average only when it is absolutely constant.

3. Nov 3, 2014

### Aristotle

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?

4. Nov 3, 2014

### haruspex

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.

5. Nov 3, 2014

### elegysix

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.

6. Nov 3, 2014

### Aristotle

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 wouldnt 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.

Thanks!

7. Nov 3, 2014

### elegysix

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?

8. Nov 3, 2014

### Aristotle

Thanks you're the best!

9. Nov 3, 2014

### haruspex

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).