What Is the Average Velocity of a Body Moving in a Circle with Constant Speed?

[bobie]

Homework Statement

I am just practicing and I found this problem:

A Body is moving with constant speed v along a circle of radius R. Find the average velocity of the body from time t=0 to t = R/3V.

Before I try to solve it, can you explain what it means?
If speed v is constant, isn't average v always just = v?

Thanks
btw: the answer is 3/pi v

 

[matineesuxxx]

Find the average velocity...If speed v is constant, isn't average v always just = v?...

Remember that velocity is different from speed. Even though a particle may be traveeling at a constant speed around a circle, its velocity is constantly changing - because its always changing direction.

Furthermore, the magnitutde of average velocity only equals average speed when the displacement is equal to the distance travelled.

 

[bobie]

Even though a particle may be traveling at a constant speed around a circle, its velocity is constantly changing - because its always changing direction.

Furthermore, the magnitutde of average velocity only equals average speed when the displacement is equal to the distance travelled.

That is really interesting, matinee, but, please, tell me:

-v is constantly changing + or -? isn't the change of v always the same as curvature (the change of direction) is always the same?

- if the body travels for a radians, isn't the displacement = r? if not, how do you calculate it?

Does this all mean that the average velocity is different if t is t0 to t=R/3v or twice as much?

 

[Nathanael]

- if the body travels for a radians, isn't the displacement = r? if not, how do you calculate it?

Displacement has nothing to do with the path taken. Displacement is simply "the difference between the final and initial positions"

So if the body travels one radian? The displacement will be a bit less than r
(it will be the length of the straight or "secant" line, not the length of the arc)"Average velocity" (defined over a specific time interval) is just the total displacement divided by the total length of time.

For example, if you went around pi radians, in one second, your average velocity would 2r m/s (if r is measured in meters)
P.S.

-v is constantly changing + or -?

Are you asking if the velocity is decreasing or increasing? The answer is neither, the magnitude is constant. The speed is not what is changing, the direction is the "part" that is constantly changing.

 

[matineesuxxx]

- if the body travels for a radians, isn't the displacement = r? if not, how do you calculate it?

If you know the radius, and the angle \theta then you can calculate x with x = 2r\sin \frac{\theta}{2}. So if a particle travels an angle theta along the arc, then its displacement is x.

 

[bobie]

, if you went around pi radians, in one second, your average velocity would 2r m/s (if r is measured in meters)

Does that mean that if I travel 2pi average v is 0? that is cool!
Now in this example, the result is 3/pi , that means that v is not relevant and the average is .95493.*2 =1.9099
Let's say that R = 6m and v =2, time is 6/3*2 = 1 sec , right?
In 1 second B travels 2m, which is 1/3 R , radians : 19.1°
distance d is 2*sin 9°.549 = .33179 *6= 1.99075 /v = 0.99537
That is not 3/pi

 

[Nathanael]

Does that mean that if I travel 2pi average v is 0? that is cool!i

Exactly!

Let's say that R = 6m and v =2, time is 6/3*2 = 1 sec , right?

6/(2*3) = 1
but
6/3*2 = 4
(parenthesis are important)

distance d is 2*sin 9°.549 = .33179 *6= 1.99075 /v = 0.99537
That is not 3/pi

Well let's not use a specific value for r (you were using 6 for that calculation right?) let's just call it "r"

So you get 0.33179*r is the displacement. But that's only the displacement, not the average speed.

 

[bobie]

So you get 0.33179*r is the displacement. But that's only the displacement, not the average speed.

I have done d/v .33*6/2 = 1.99075 that's the average
but it should be 1.99099
Do you think the result given by OP 3/pi may be wrong?

 

[Nathanael]

I have done d/v .33*6/2 = 1.99075 that's the average
but it should be 1.99099
Do you think the result given by OP 3/pi may be wrong?

Oh, sorry about that. Yes I think it is.

 

[bobie]

Remember that velocity is different from speed. Even though a particle may be traveeling at a constant speed around a circle, its velocity is constantly changing - because its always changing direction.

Can you guys tell me if this concept of average v in circular motion is of any practical use.
Can you mention one instance when it comes in handy?

Thanks

 

[ehild]

I think the only practical use is to make it as a problem in tests, to confuse students.

ehild

 

[matineesuxxx]

I think the only practical use is to make it as a problem in tests, to confuse students.

ehild

I totally agree.

This is one of those types of questions that you may find on a quiz or exam to show them that you truly understand the concept, where you need to be confident in your work and not start second guessing yourself.

 

[bobie]

Thanks, a sigh of relief!

 

[HallsofIvy]

Can you guys tell me if this concept of average v in circular motion is of any practical use.
Can you mention one instance when it comes in handy?

Thanks

Perhaps to see if you really understand the concept of "average speed". If you are going around in circles you are not going anywhere so of course your average speed is 0.

 

[ehild]

Perhaps to see if you really understand the concept of "average speed". If you are going around in circles you are not going anywhere so of course your average speed is 0.

You mixed speed and velocity I am afraid. It is the average velocity that is zero if you go a full circle.

ehild

 

[olivermsun]

I think the only practical use is to make it as a problem in tests, to confuse students.

ehild

Actually, the concept of a vector average of "something" in an oscillatory system is used everywhere in math, physics, engineering, …

 

[ehild]

Actually, the concept of a vector average of "something" in an oscillatory system is used everywhere in math, physics, engineering, …

As for velocity, it is usually in the context that the average is zero.


ehild

 

[bobie]

I think the only practical use is to make it as a problem in tests, to confuse students.

As for velocity, it is usually in the context that the average is zero

Probably this concept is useful to establish that a ball rotating on a string has p = 0 (m * v = 0) ?

 

[ehild]

Probably this concept is useful to establish that a ball rotating on a string has p = 0 (m * v = 0) ?

The ball has velocity at every instant so its momentum is not zero. But the time average for a full period is zero.

ehild

 

[olivermsun]

As for velocity, it is usually in the context that the average is zero.

Yes, or close to it.

As you know, we're often interested in things that don't quite make it to zero. For example: average the velocity of a certain bit of water (or suspended particle) in a surface wave on the ocean.