# Measuring parallel velocity in rotating frame.

1. Feb 11, 2014

### peripatein

1. The problem statement, all variables and given/known data
At t=0 a police officer is located at (0,R) on a circular platform whose radius is R and which rotates around the z axis with constant angular velocity ω. The officer's velocity at that point in time is (ωR ,0). At that time, a bird leaves the center of the platform along the x axis with constant velocity (v0,0). The officer is equipped with a laser velocity-meter which measures the velocity parallel to the beam. The officer aims the device at the bird. What would be the velocity of the bird as detected by the officer?

2. Relevant equations

3. The attempt at a solution
The officer's velocity should be (I think):
Vofficer = ωR(-sin(ωt),cos(ωt)). However, I am not sure what the bird's velocity would be. In the lab's reference frame the bird would obviously be moving in a straight line. Moreover, if in the rotating reference frame its velocity is constant then wouldn't its velocity simply be: v0(-sin(ωt),cos(ωt))? I am quite sure this is wrong, but I am making an effort. I'd appreciate some help with this.

2. Feb 11, 2014

### voko

Is your velocity formula for the officer compatible with the initial conditions?

Regarding the rest of the problem, both velocities are vectors. What is the bird's velocity relative to the officer?

3. Feb 11, 2014

### peripatein

Right, the officer's velocity should be:
ωR(cos(ωt),sin(ωt))
Relative to the officer, the bird is flying with additional v0 along the x-axis, isn't it?

4. Feb 11, 2014

### voko

Where should the officer be when $\omega t = \pi / 2$? Where should the velocity be directed? Is that compatible with your formula?

The bird is flying along the x-axis in the lab frame. The officer is measuring velocity in the officer's frame. How can you reconcile the two?

5. Feb 11, 2014

### peripatein

Let's focus on the bird for a second. Silly me, for some peculiar reason I figured it would be moving along the x-axis in the rotating frame instead of flying in the lab. In any case, the reconciliation should be via:
vrotating=vlab - ωXr (all these are vectors of course).
After substituting the appropriate relations for the unit vectors as well, I obtained:
vbird=(v0cos(ωt)-v0ωtsin(ωt),-v0sin(ωt)-v0ωtcos(ωt)) in the rotating frame.
Does that seem correct to you?

6. Feb 11, 2014

### voko

No, that does not seem correct. Especially because the bird's velocity is growing unbounded over time.

Is there any reason why you have to convert the bird's velocity to the rotating frame?

7. Feb 11, 2014

### peripatein

Would it be better to leave it as v0 in the lab's RF and convert the officer's velocity to the lab's RF?

8. Feb 11, 2014

### voko

You don't have to convert the officer's velocity to the lab frame. You only have to get it right, as I suggested in #2 and #3.

9. Feb 11, 2014

### peripatein

The platform rotates clockwise (not counter-clockwise). Hence, at y=R and t=0, the angle should be zero I think, and not pi/2. Therefore, the officer's velocity should be:
ωR(cos(ωt),-sin(ωt))
Is this correct?

10. Feb 11, 2014

### voko

That looks good to me.

11. Feb 11, 2014

### peripatein

Okay, now is this the officer's velocity in the lab reference frame?
Also, how do I proceed? Do I now need to subtract this velocity from the bird's and multiply by the cosine of the angle between the two velocities (to obtain the velocity parallel to the officer's)?

12. Feb 12, 2014

### voko

Why do you need a "velocity parallel to the officer's"? The officer's device measures "the velocity parallel to the beam". What is "the beam" here?

13. Feb 12, 2014

### peripatein

Is the beam tantamount to the bird?

14. Feb 12, 2014

### peripatein

Would you be so kind as to at least list the steps necessary for solving this problem, having determined the officer's velocity?

15. Feb 12, 2014

### peripatein

I know the final answer, but am not sure how to obtain it. The final answer is:
{v02t-v0R[sin(ωt)+ωtcos(ωt)]}/√[R2+(v0t)2-2v0Rtsin(ωt)]
I don't have much time. Am doing this for an exam I shall have to take within a couple of hours and after the exam it will be far less relevant. If you could guide me through by listing all the necessary steps I'd appreciate it.

16. Feb 12, 2014

### voko

The "beam" is the line of sight from the officer to the bird. You need to obtain the projection of the bird's relative velocity (relative to the officer's velocity) onto the beam.

17. Feb 12, 2014

### peripatein

Do I need to find the vector connecting the officer and the bird then? And then project the relative velocity onto that vector?

18. Feb 12, 2014

### peripatein

The bird's position is given by v0t, whereas the officer's could be obtained by integrating his velocity vector, couldn't it?

19. Feb 12, 2014

### voko

Yes that is correct. Even though you should have been able to guess the officer's position from the data given.

20. Feb 12, 2014

### peripatein

Alright, so upon finding the vector connecting the two, do I simply multiply the bird's velocity vector by the cosine of the angle between the former vector and the latter (that is, the angle between the beam and the bird's velocity)?