Graph of Position Seems to be Wrong

[GoCubs12]

Homework Statement

By relative velocity, we mean velocity with respect to a specified coordinate system. (The term velocity, alone, is understood to be relative to the observer’s coordinate system.)

(a) A point is observed to have velocity Va relative to coordinate system A.What is its velocity relative to coordinate system B, which is displaced from system A by distance R? (R can change in time.)

(b) Particles a and b move in opposite directions around a circle with angular speed ω, as shown. At t = 0 they are both at the point r = lˆj, where l is the radius of the circle. Find the velocity of a relative to b.

(c) Sketch the motion of the particle.

Homework Equations

Theta=wt

The Attempt at a Solution

I was able to solve part A and found Va=Vb + dR/dt. I'm pretty confident in this answer.

I was also able to solve part B. I found dR/dt to be 2wlcos(wt). I then plugged this into the answer from part A and got Va=Vb + 2wlcos(wt). I'm pretty confident in this answer as well.

My issue is with part C. To graph the position, I integrated the answer from part B and got Rx=Vbt+2lsin(wt). I believe I did this right, but if you graph it the sin function does not go linearly across the graph as it normally does, but instead gradually increases due to the Vbt term in front of it. This does not seem to make sense in the real world because the particles are going around a circle so they should not be gradually getting farther apart over time? I'm pretty stumped on this. Any input about this would be appreciated.

Thanks for all the help!

 

[Stephen Tashi]

My issue is with part C. To graph the position, I integrated the answer from part B and got Rx=Vbt+2lsin(wt). I believe I did this right, but if you graph it the sin function does not go linearly across the graph as it normally does, but instead gradually increases due to the Vbt term in front of it.

If we are assuming particle B's coordinate system places (at all times) particle B at the origin of that system, then particle B's velocity in that system is zero.

 

[GoCubs12]

If we are assuming particle B's coordinate system places (at all times) particle B at the origin of that system, then particle B's velocity in that system is zero.

So since particle B's position would always be (0,0), it could just be left out of all the equations then? Including in part A?

 

[Stephen Tashi]

So since particle B's position would always be (0,0), it could just be left out of all the equations then? Including in part A?

No. In part a), I interpret the phrase "coordinate system B, which is displaced from system A by distance R" to mean the origin of coordinate system B is displaced from the origin of coordinate system A by displacement vector R. That does not imply that when we do computations of a particle's velocity that we always assume the particle is at the origin of one of the coordinate systems.

In the special case where a problem asks about something "relative to particle A", it usually means we do use a coordinate system where the particle is at the origin of "its" coordinate system.

 

[GoCubs12]

No. In part a), I interpret the phrase "coordinate system B, which is displaced from system A by distance R" to mean the origin of coordinate system B is displaced from the origin of coordinate system A by displacement vector R. That does not imply that when we do computations of a particle's velocity that we always assume the particle is at the origin of one of the coordinate systems.

In the special case where a problem asks about something "relative to particle A", it usually means we do use a coordinate system where the particle is at the origin of "its" coordinate system.

Thanks that helps a lot. Also, in part B it is talking about being relative to particle B so the Vb term could be left out since it would imply that it is 0?

 

[Stephen Tashi]

Thanks that helps a lot. Also, in part B it is talking about being relative to particle B so the Vb term could be left out since it would imply that it is 0?

The way I'd phrase it is that the Vb is zero so the Vb term has no effect. "Left out" implies something about notation, which may or may not get the approval of a grader.

 

[GoCubs12]

The way I'd phrase it is that the Vb is zero so the Vb term has no effect. "Left out" implies something about notation, which may or may not get the approval of a grader.

Thanks for all the help!