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

1. Aug 18, 2015

### Dennis Heerlein

• Missing homework template due to originally being posted in other forum.
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?

2. Aug 18, 2015

### Orodruin

Staff Emeritus
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.

3. Aug 18, 2015

### Dennis Heerlein

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

4. Aug 18, 2015

### Orodruin

Staff Emeritus
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.

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?

5. Aug 18, 2015

### Dennis Heerlein

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

6. Aug 18, 2015

### Orodruin

Staff Emeritus
This of course depends on the actual speed with which you are travelling. But let us say you are travelling 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.

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

7. Aug 18, 2015

### andrewkirk

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
$$\frac{\int_{t1}^{t2} \|\vec{v}\|dt}{t2-t1}$$

8. Aug 18, 2015

### 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?

9. Aug 19, 2015

### haruspex

Average speed.