A high school physics problem demonstrating relative motion

[wrobel]

I remembered a pretty high school problem from kinematics. But it seems it can help even undergraduates to develop their understanding of what a relative motion is.
Consider a railway circle of radius $r$. Assume that a carriage running along this circle has a speed $v$. See the picture. A fly $M$ flies in the opposite direction and has a speed $u,\quad |OM|=b$. Find a speed of the fly relative to the carriage.
The obvious incorrect answer is $u+v$ while the correct answer is $u+bv/r$.
The point is as follows. The velocity of any point is defined relative to a frame. To say "velocity relative to the carriage" is the same as to say "velocity relative to a frame connected with the carriage" Thus in this problem the frame rotates about the point $O$ with the angular velocity $v/r$.

 

[Andrew Mason]

I am not clear on the facts. Is the fly prescribing circular motion about O in the clockwise direction?

In any event, velocity of the fly relative to the carriage frame would just be:
$\vec{u}-\vec{v}$ where the velocity vectors $\vec{v}\text{ and } \vec{u}$ are constant magnitude but one or both constantly changing direction.

At the moment that they are both positioned as shown, the relative motion of the fly to the carriage would just be $|u|\hat{-j}-|v|\hat{j}=-(|u|+|v|)\hat{j}$ where $\hat{j}$ is the unit vector in the page-up direction.

AM

 

[haruspex]

I am not clear on the facts. Is the fly prescribing circular motion about O in the clockwise direction?

In any event, velocity of the fly relative to the carriage frame would just be:
$\vec{u}-\vec{v}$ where the velocity vectors $\vec{v}\text{ and } \vec{u}$ are constant magnitude but one or both constantly changing direction.

At the moment that they are both positioned as shown, the relative motion of the fly to the carriage would just be $|u|\hat{-j}-|v|\hat{j}=-(|u|+|v|)\hat{j}$ where $\hat{j}$ is the unit vector in the page-up direction.

AM

The question is the speed of the fly as perceived by an observer in the carriage. We can simplify it by first making the fly stationary and having the carriage move in a straight line. Now the answer is v. But then add that the observer is turning around within the carriage at angular rate v/r. That adds (v/r)(b-r) to yield vb/r.

 

[wrobel]

See Velocity addition theorem

 

[Andrew Mason]

The question is the speed of the fly as perceived by an observer in the carriage. We can simplify it by first making the fly stationary and having the carriage move in a straight line. Now the answer is v. But then add that the observer is turning around within the carriage at angular rate v/r. That adds (v/r)(b-r) to yield vb/r.

??? The velocity of the carriage relative to the fly is the time rate of change of the displacement vector of the carriage to the fly. In this diagram, $\vec{v_{c-f}}=\dot{\vec{FC}}$:

But $\vec{FC}=\vec{R}-\vec{OF}$ so $\vec{v_{c-f}}=\dot{\vec{FC}}=\dot{\vec{R}}-\dot{\vec{OF}}=\dot{\vec{R}}$

AM

 

[haruspex]

??? The velocity of the carriage relative to the fly is the time rate of change of the displacement vector of the carriage to the fly. In this diagram, $\vec{v_{c-f}}=\dot{\vec{FC}}$:

View attachment 369191

But $\vec{FC}=\vec{R}-\vec{OF}$ so $\vec{v_{c-f}}=\dot{\vec{FC}}=\dot{\vec{R}}-\dot{\vec{OF}}=\dot{\vec{R}}$

AM

Well, this is interesting.

At e.g. https://en.wikipedia.org/wiki/Relative_velocity I read
"The relative velocity of an object B with respect to an observer A, is the velocityvector of B measured in the rest frame of A"
But how is rest frame defined? Is it the frame fixed to the rigid body, or the inertial frame in which the mass centre is instantaneously at rest, or the non-rotating frame fixed to its mass centre? Following the wikipedia link didn't help since it referred only to particles.
@wrobel is clearly using the first interpretation, and I presumed he was right, whereas I am seeing the last on the web.

 

[Andrew Mason]

Well, this is interesting.

At e.g. https://en.wikipedia.org/wiki/Relative_velocity I read
"The relative velocity of an object B with respect to an observer A, is the velocityvector of B measured in the rest frame of A"
But how is rest frame defined? Is it the frame fixed to the rigid body, or the inertial frame in which the mass centre is instantaneously at rest, or the non-rotating frame fixed to its mass centre? Following the wikipedia link didn't help since it referred only to particles.
@wrobel is clearly using the first interpretation, and I presumed he was right, whereas I am seeing the last on the web.

Those definitions are a bit confusing as they use velocity in the definition of relative velocity.

I would suggest that a better and less confusing definition of velocity of A relative to B at time $t$ is the time derivative at time $t$ of the displacement of A relative to B in the inertial frame attached to B at time $t$. In other words, it is the limit of the the change in displacement vector of A to B in that frame over a time interval from $t$ to $t+\Delta t$ divided by that time interval as $\Delta t \rightarrow 0$:

$\vec{v_{A-B}}=\lim_{\Delta t\rightarrow 0}\frac{\Delta\vec{AB}}{\Delta t}$

It is apparent that A's relative velocity to B's inertial reference frame at time $t$ is the same for all points in that inertial frame, as the displacement of any point relative to another point in the same reference frame is constant.

If the second time derivative of that displacement vector is non-zero, that just means that A is undergoing acceleration relative to B. That does not tell you which are accelerating relative to an inertial point. To determine that, one just has to use any inertial point as a reference.

AM