Where did I go wrong? (an object moves through a half circle)

[Dennis Heerlein]

This is the question:
During a time interval of 5 seconds, an object moves through a half circle with a radius of 10 meters, as shown above. What is the magnitude of the object's velocity during this motion?

Note: I do not know how to post the diagram, but it simply shows a curved arrow going around a half circle, with the object ending exactly across from its starting point.
I solved this using V=2(pi)(r)/Period.
The period of half of a circle is five seconds so the period of a full circle is ten seconds.
So: 2pi(10)/(10 seconds)
I got an answer of 2 pi m/s.
The book says the answer is 4 m/s because the change in displacement is twenty meters (diameter) and the time is five seconds, so d/t=20/5=4 m/s.

Who went wrong?

 

[Orodruin]

During a time interval of 5 seconds, an object moves through a half circle with a radius of 10 meters, as shown above. What is the magnitude of the object's velocity during this motion?

Is this the problem statement exactly as stated? The book is computing the magnitude of the average velocity, which is not necessarily equal to the average of the magnitude of the velocity.

 

[Dennis Heerlein]

This is exactly as stated, word for word. So can you explain who is wrong, and the difference of those you stated?

 

[Orodruin]

So can you explain who is wrong, and the difference of those you stated?

No, the question is ambiguous. It is not clear whether it is the magnitude of the average velocity or the average of the magnitude of the velocity which is intended. The author seems to have intended the former, but just reading the statement I might interpret it as the latter as well. It is simply a badly posed question. If the author had written "find the magnitude of the average velocity" or "find the average speed", the problem would have been fine.

and the difference of those you stated?

If you go around the full circle at constant speed (which is the magnitude of the velocity) and end up where you started, what would have been your average velocity (remember, velocity is displacement/time so average velocity is total displacement / total time)? What would have been your average speed?

 

[Dennis Heerlein]

Average speed would be 2pi m/s? Average velocity would be 4m/s?

 

[Orodruin]

Average speed would be 2pi m/s?

This of course depends on the actual speed with which you are travelling. But let us say you are traveling constantly at 1 m/s - then your average speed will be ... 1 m/s. Nothing strange here, the time average of a constant quantity is the quantity itself.

Average velocity would be 4m/s?

If you go a full lap around the circle (which is what I was asking about), what is your total displacement?

 

[andrewkirk]

Average speed would be 2pi m/s? Average velocity would be 4m/s?

No, the average velocity will be zero, because that's net displacement divided by time and net displacement is zero. Average speed will be just whatever the constant speed was. In symbols, the magnitude of average velocity is
$$\frac{\|\int_{t1}^{t2} \vec{v}\,dt\|}{t2-t1}$$
whereas the average speed is
$$\frac{\int_{t1}^{t2} \|\vec{v}\|dt}{t2-t1}$$

 

[Dennis Heerlein]

Okay. If a full lap around the circle is made then the average velocity would be zero due to zero displacement. But what does the formula 2pir/T measure then?

 

[haruspex]

Okay. If a full lap around the circle is made then the average velocity would be zero due to zero displacement. But what does the formula 2pir/T measure then?

Average speed.