B How does the Gallilean Invariance Puzzle challenge our understanding of motion?

[Jeff Root]

It looks to me that what led the original poster astray was
the math. The situation is simple and intuitively obvious.
Applying the math to that situation without understanding
exactly how the math connects to it is the problem. If the
situation is described in one frame of reference only, the
problem does not arise. Only when one tries to work out
mathematical descriptions of both frames together do the
descriptions begin to seem contradictory.

The math is often essential to work out what happens,
especially when a quantitative result is needed, but it can
also be counterproductive to understanding what happens.

Ooo! You brought to mind something from a college class.
Public speaking or some such. It was a demonstration of
persuasion, and it happened to work out such that the impact
on me must have been greater than on anyone else in the
class of about 20 students. We were told this story, as best
I can recall now:

A farmer bought a calf for $30. A few weeks later he sold it
for $40. A few weeks after that he bought it back for $50.
Then a few weeks later he sold it again for $60. A few weeks
later yet he bought it back again for $70, and then after a
few weeks more he sold it again for $80.

How much money did the farmer make or lose?

We were not told not to write anything down, or do any
calculations. (This was just before the first pocket calculators
came out.) That felt like cheating, though, so I didn't do it.
I didn't notice anyone else writing or doing calculations.

Students volunteered answers. I believe that five different
answers were put forward. I raised my hand and offered
mine: There were three iterations, and the farmer made
$10 each time, so I figured he made $30. Others suggested
other amounts ranging from $10 on up to maybe $80.
I wasn't the first to give an opinion; I may have been the last.

I think we assumed zero costs for feed. I don't recall that
being specified. Maybe the calf just ate grass.

We were asked to group together with whoever expressed
a money amount we agreed with. I remember that there were
five different answers because I clearly remember one group
in each corner of the room and one in the center. I think three
others joined the group I started. Two gals and a guy.

The sizes of the groups were remarkably uniform: About four
people in each of the five groups. I still think that is interesting.

We discussed the problem for a few minutes. The other guy
who joined my group said he was an accounting major. He
started explaining the math. He went through the process
and came up with a different number than I had. He went
through it again, and by the time the instructor told us to
quit, all four of us appeared to be convinced that my guess
was mistaken. He went to the chalkboard and tried to show
the whole class his analysis. But something went wrong.
The numbers didn't work out. Of course it turned out in the
end that -- to my astonishment -- my original rather reckless
guess was right, and the accounting major's math was wrong.
I was persuaded by a good-sounding argument in which the
math seemed to work, but actually didn't. My analysis was
correct, that the farmer made $10 on each transaction, and
over three transactions he made $30. But I was persuaded
that I had made a mistake. I am not proficient at math, even
simple arithmetic, and I knew it. I was not surprised to be
shown I was wrong. I was very surprised to learn that I was
(apparently) the only one in the class who had been right.

As an example of one way persuasion can work, that exercise
has to have impressed me more than anyone else.

-- Jeff, in Minneapolis

 

[jefferywinkler2]

Let’s say you were standing on the ground, and a wall was moving towards you, and you threw a rubber ball at the wall which bounced off. The wall moving towards you would transfer some of its momentum to the ball so the ball would come back towards you with higher velocity than when you threw it.

delta p_wall = delta_ball
p = mv
m_wall delta v_wall = m_ball delta v_ball

Both m_wall and m_ball are fixed so an increase in velocity of the ball results in a decrease in velocity of the wall.

Now if imagine you are on a moving train which is passing a wall perpendicular to the train. The wall is stationary with respect to the ground. As the train approaches the wall. you throw the ball out the window to the wall, and the ball bounces off the wall. Assume that the angle between the path of the ball and the track is negligible. From the reference frame of the train, it is identical to the earlier example, where you are motionless, and the wall is moving towards you.

delta p_wall = delta_ball
p = mv
m_wall delta v_wall = m_ball delta v_ball

Both m_wall and m_ball are fixed so an increase in velocity of the ball results in a decrease in velocity of the wall.

However, the wall is only moving from the reference frame of the train. In the reference frame of the ground, the wall is stationary, and the train is moving. In order for the wall to slow down in the reference frame of the train, the train would have to slow down in the reference frame of the ground.

Does the ball bouncing off the wall cause the train to slow down?

 

[Ibix]

Does the ball bouncing off the wall cause the train to slow down?

I think you are describing a scenario where you have a wall stationary beside the track and your train is approaching it. You lean out of the window and throw a ball at the wall and catch it.

Let's call momentum positive when it is directed towards the front of the train. There are three interactions.

First you throw the ball. Its momentum increases and the train's decreases.

Second the ball bounces off the wall. The ball's momentum decreases and the wall's increases.

Third you catch the ball. Its momentum increases and the train's decreases.

So yes, the train is slowed by this process, both when you throw and when you catch the ball. Note that I haven't specified the frame I'm working in, and I can get away with this because the momentum changes are the same in any one.

 

[A.T.]

Does the ball bouncing off the wall cause the train to slow down?

If you consider wall+earth as movable and ignore the train's propulsion, then the relative velocity between ground and train will decrease, when the ball hits the wall. But due to the mass ratio the wall+earth are usually considered immovable in a collision with a ball.

 

[A.T.]

So yes, the train is slowed by this process, both when you throw and when you catch the ball.

And when the ball bounces from the wall, the train's groundspeed also decreases, because the wall accelerates forward.

Note that I haven't specified the frame I'm working in, and I can get away with this because the momentum changes are the same in any one.

In any inertial one.