Instantaneous Speed = Instantaneous Velocity?

[in the rye]

Hi everyone,

I am taking my first physics class (algebra based), and yesterday we were covering 1-Dimensional motion. I'm enrolled in Cal. 2, so I have some understanding of instantaneous velocity, etc. However, my professor said that instantaneous speed is always equal too instantaneous velocity, while average speed is only ever equal to the magnitude of the velocity vector, so long as there is no change in direction.

The book we're going through did not explain this, so I'm having trouble understanding WHY, but let me test if my logic is correct.

Average speed = magnitude of the velocity vector, so long as there is no change in direction, because if the direction changes, we subtract distance from it since we are only concerned about our displacement. However, with instantaneous velocity since our change in time is infinitesimally small, there is technically never any change in direction that time interval. So, essentially, the difference between instantaneous speed, and instantaneous velocity is solely that our velocity will be a vector (have direction) at that point, whereas instantaneous speed will only be a scalar.

Is this right?

And unrelated, are there any books that would be good for just explaining physics conceptually? I'm comfortable reading them if they are calculus based. My book I currently have just runs through example, after example, without really getting into the concepts. My professor, however, teaches mostly concepts. So I get a good mix. I'd just like something that can reiterate his points.

 

[HallsofIvy]

I think you are misunderstanding "instantaneous". You say "with instantaneous velocity since our change in time is infinitesimally small, there is technically never any change in direction that time interval." There is NO "time interval" with "instantaneous". But we can have an instantaneous change in direction as well as in numerical value.

 

[Dale]

Are you familiar with uniform circular motion yet? It may be the easiest way to show the differences.

In uniform circular motion you can write your instantaneous velocity (in some units) as ##v=(-\sin(t),\cos(t))##. Your instantaneous speed is the magnitude of ##v##, so if you work that out then you find it is a constant speed of ##1##.

If you average the velocity from ##t=0## to ##t=2\pi## then you get an average velocity of ##(0,0)## which makes sense because you came back to where you started. If you average the speed over the same time then you get ##1##, which also makes sense because you never sped up or slowed down you just turned.

 

[in the rye]

I think you are misunderstanding "instantaneous". You say "with instantaneous velocity since our change in time is infinitesimally small, there is technically never any change in direction that time interval." There is NO "time interval" with "instantaneous". But we can have an instantaneous change in direction as well as in numerical value.

I understand instantaneous as being a limit as your time interval approaches 0, hence why I qualified it as being infinitesimally small, that is, it is so small it is essentially 0.

Are you familiar with uniform circular motion yet? It may be the easiest way to show the differences.

In uniform circular motion you can write your instantaneous velocity (in some units) as ##v=(-\sin(t),\cos(t))##. Your instantaneous speed is the magnitude of ##v##, so if you work that out then you find it is a constant speed of ##1##.

If you average the velocity from ##t=0## to ##t=2\pi## then you get an average velocity of ##(0,0)## which makes sense because you came back to where you started. If you average the speed over the same time then you get ##1##, which also makes sense because you never sped up or slowed down you just turned.

I've not covered uniform circular motion, so I don't quite see it from your example.

I think I'm just trying to understand it in the conceptual nature, rather than its applicability because once I grasp the concept, the applications come naturally to me. I don't know if this is typical in physics, however. It seems like the labs, etc. kind of help with the concepts. But, it still isn't making sense.

I guess what I'm analyzing is this idea that if our time interval is so small that it technically doesn't exist, then the velocity doesn't even have a chance to turn around and head back the opposite direction. Which is why it would be equal to the speed. Whereas, if we have a large time interval it could go all over the place. Putting it into an example, if we a race car on a race track, we could sample its instantaneous speed and instantaneous velocity, and they would be equal. However, if we take their average speed, and average velocity over the entire race, our average speed would be the total distance/total time, but our velocity would be 0 since there is no displacement from start to finish (assuming they ended in the same exact position). This would be due to the large time interval. Am I way off base?

That being said is it instantaneous speed is only equal to the magnitude of v or is it equal to v.

 

[ehild]

In one dimension, instantaneous velocity can be either positive and negative. It is a one-dimensional vector. Instantaneous speed is a scalar, magnitude of the instantaneous velocity, and can not be negative. They are different concepts.

 

[PeroK]

I understand instantaneous as being a limit as your time interval approaches 0, hence why I qualified it as being infinitesimally small, that is, it is so small it is essentially 0.
I've not covered uniform circular motion, so I don't quite see it from your example.

I think I'm just trying to understand it in the conceptual nature, rather than its applicability because once I grasp the concept, the applications come naturally to me. I don't know if this is typical in physics, however. It seems like the labs, etc. kind of help with the concepts. But, it still isn't making sense.

I guess what I'm analyzing is this idea that if our time interval is so small that it technically doesn't exist, then the velocity doesn't even have a chance to turn around and head back the opposite direction. Which is why it would be equal to the speed. Whereas, if we have a large time interval it could go all over the place. Putting it into an example, if we a race car on a race track, we could sample its instantaneous speed and instantaneous velocity, and they would be equal. However, if we take their average speed, and average velocity over the entire race, our average speed would be the total distance/total time, but our velocity would be 0 since there is no displacement from start to finish (assuming they ended in the same exact position). This would be due to the large time interval. Am I way off base?

That being said is it instantaneous speed is only equal to the magnitude of v or is it equal to v.

Maybe look at this a different way. Instantaneous velocity means the velocity at a single point/instant in time. [IF you are not sure that such a thing exists, then you may need to start thinking in terms of limits or "infinitesimal" time intervals. But, when I think about physics, I have no trouble with the idea that a particle has a velocity at each point in time.]

So, I'm quite happy with the concept that at each instant a particle has a velocity. At that instant, its speed is just the magnitude of the velocity. Technically, therefore, as already pointed out in post #5: speed = +/- velocity (for 1 dimensional motion).

More generally, in terms of vectors: ##v = |\vec{v}|##, where ##v## is the speed and ##\vec{v}## is the velocity.

Now, there is also the concept of "average" speed and "average" velocity. The average velocity (over a period of time) is the displacement divided by the time interval. As you probably know, if you move out and then back to where you started, the average velocity is 0.

The average speed, however, is the total distance traveled divided by the time interval. The average speed can never by 0 unless you haven't moved at all.

Average speed and average velocity, therefore, often bear little relation to each other. Especially in 2 and 3 dimensional motion.

 

[in the rye]

Right, gotcha. That's because by definition magnitude is always positive. The +/- is only indicative of the direction on the coordinate system.

 

[CWatters]

You get a similar issue in electronics. If you look at an AC voltage waveform the average voltage is zero - so you might be fooled into thinking the average power to be zero but that's not the case.