# Question

This is from my textbook:

There are 2 balls of the same size made of rubber and iron respectively kept on the smooth surface of a moving train. Which ball will move faster when the train brakes suddenly?

In that start it seems simple. Let the iron ball be called $A$ and the rubber one $B$. The mass of the iron ball is greater as:

$$(1)\ M = V \cdot D \\ (2)\ D_A > D_B \\ (3)\ V_A = V_B \ \ [From\ Q] \\ (4) M_A > M_B \ \ [From (1), (2), (3)] \\$$

Then we have to consider what forces are acting on it. The only ones I could think of were friction with the floor and air resistance.

If we consider friction, then we can easily conclude that the rubber ball moved slower as rubber creates a lot of friction. But since they mentioned the floor was 'smooth', I thought we should ignore that.

So all that's left is air resistance (denoted by $F$).

$$(5) F = m \cdot a \ \ \rightarrow \ \ a = \frac{F}{M} \\ (6) Surface\ Area_A = Surface\ Area_B \ \ [From (3)] \\ (7) F_{iron} = F_{rubber} \ \ [From (6)] \\ (8) a_A < a_B \ \ [From (4),(5),(7)]$$

That means the rubber ball will move faster than the iron one (and therefore that the iron ball will stop before the rubber one does).

But my teacher (who'd given me this problem) disagrees. Her argument is:

Consider everything with reference to the train. Before braking, the balls are stationary (w.r.t. the train). When the train brakes, negative acceleration is provided to the train. That means the balls will move forward w.r.t. the train. But since the iron ball has a greater mass and so inertia, it will move slower.

### Doubts

So my questions are:

1. Was my teacher's approach correct? Seems a bit unsound to me, but that just may be b'coz I don't have enough intuition for switching inertial frames of reference.

2. Are there any other forces in play over here? I could only think of 2: air resistance and friction.

3. Should air resistance be considered?

4. Should friction be considered? In that case, the rubber ball will be significantly slowed.

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Did your teacher consider air resistance (or any other force like friction force) or not? – Trimok Jul 22 '13 at 8:51
All that she told me, I have written. Now my hols are going on, so I can't ask either. Though I can bet she wasn't even thinking along the lines of a force being applied resulting in acceleration – YatharthROCK Jul 22 '13 at 8:59
@Trimok Are there any other forces that can be considered? I could only think of 2... – YatharthROCK Jul 22 '13 at 9:00
Since the question says "kept on the smooth surface", you're probably supposed to assume there are no forces acting on the balls at all. – Nathaniel Jul 22 '13 at 9:02
Consider a "rest" frame, in which the initial speed of the train and the balls is $v$. When the train is stopping, and if there is no force exerced on the balls, these balls go on at the same speed $v$ (this is the 1st Newton's law of motion, whatever the movement of the train is. When the train has stopped, it becomes a "rest" frame too (the speed of the stopped train, relatively to the original "rest" frame, is obviously zero), so the speed of the balls, relatively to the stopped train frame, is the same, and is equal to $v$, too. – Trimok Jul 22 '13 at 9:15

If you ignore air resistance, the answer is "neither of them". There are no forces acting on either of the balls, so they will keep on moving at the same speed the train was moving originally, until they roll (or rather slide, if there is no friction at all) off the surface or hit something. In the frame of the train, they both receive the same acceleration, so they will both move in exactly the same way. This is the same reason why (neglecting air resistance) the two balls would fall at the same speed if dropped from a tall building.

The question probably wants you to ignore air resistance, but in case you're interested in how air resistance would affect the balls' motion, here are some details. It is again very similar to if they were dropped: they would both start out accelerating at the same rate (relative to the train), but the heavy ball would slow down less rapidly, so it would ultimately go further. The reason for this is that both balls receive (more or less) the same drag force due to air resistance, but because $a=F/m$ this force causes less deceleration for the heavier ball. The balls will be moving much slower than if they were falling, so air resistance would probably not have a very large effect if you actually performed this experiment.

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I did not look at the equations (I avoid it whenever I can).

I disagree with the statement atributed to your teacher. The main reason is that when the train brakes, you do not really care about the braking acceleration, as a first approximation. In the absence of other forces, the balls will just keep going at the same speed, together. Newton was standing at the station and said so. That is also true when the train only slows down. Acceleration (or intermediate speeds) may matter when analyzing friction forces, with the air or the surface.

Now, what interaction may matter, assuming the surface and the air slows and stops with the train.

The braking force by air is the same on both (same size), for the same speed. The (braking) acceleration from that force is inversely proportional to mass, and thus it will have more braking effect on the lighter rubber ball.

The second interaction is with the surface. The hard iron ball is much more likely to skid than the rubber ball (though the lighter weight of the rubber ball may help it skid too). Skidding consumes energy from friction, and that slows more the lighter ball (though it is not the same friction force for both balls: different weight and different substances). On the other hand, not skidding imparts an angular speed to the ball. Sharing the available translational kinetic energy of the ball between rotation and translation implies that the ball slows down in translation (identical effect for both balls)

If the train brakes slowly (weak acceleration, hence weak forces, thus relative rolling speed can increase slowly) there is a chance that skidding is avoided for both balls. If rolling friction and air resistance are very small, the two balls may stay close for a long distance.

More precise analysis requires actual figures.

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I did not know what year 9 means. I am not in the USA. I hope my answer is not too complex.(forget my quip about equations) – babou Jul 22 '13 at 20:59
+1'd, 10x for the answer. I understood the answer and it was quite detailed :) But I didn't quite get your second statement: "I disagree with the statement attributed to your teacher". – YatharthROCK Jul 23 '13 at 13:33
@YatharthROCK I meant that I disagree with the idea that the iron ball is slower. There seems to be a confusion between force and acceleration. I am not used to reasonning in terms of frames, but since you do, I believe it is much simpler to take the train station as inertial frame. In that frame, no force is applied to the balls, and they just keep going at the same speed (or rather at the same velocity). If you want to consider various frictions, then there are different ways you can reason. But all this is just my reasonning biases. – babou Jul 23 '13 at 20:57
No, that was not the source of my confusion. I'd mistakenly read the statement as "I disagree with the fact that the statement was being attributed to your teacher" and not as " I disagree with the statement that you had attributed to your teacher". Sorry... – YatharthROCK Jul 24 '13 at 4:07

Answering my own doubts in order:-

1. No. It got to the correct answer, but was wrong.

I think she was getting confused between cause (force) and effect (acceleration). When the train brakes, the ball and train acquire a relative acceleration. No other force comes into play when one's inertial frame of reference is the train's frame.

She was basically saying $a_{iron} = a_{rubber}$ and $M_{iron} > M_{rubber}$, so therefore $F_{iron} > F_{rubber}$ which is correct. But then she also says that the iron ball will move a longer distance which implies that both the balls will eventually come to a stop while simultaneously claiming to ignore air resistance and friction. So then which force eventually stops the balls (as otherwise they'll keep moving at the constant speed otherwise as dictated by one of Newton's Laws of Motion)?

2. None that I could think of (which are probably all of them considering this is such a simple question).

3. Not mentioned, but probably not. In most idealized physics questions (especially textbook ones), air resistance is not something to be considered. This is to some extent implied by the phrase "on a smooth floor".

If we don't consider it, then no external forces will be acting on the ball (barring friction) and both the balls will move at the same speed till they like slide off the train's surface or hit a wall or something meaning this was a trick question (which seems atypical as it isn't the modus operandi of the crappy, unenlightening educational system).

If we do consider it, then b'coz of the reasons in the (my) original answer, the iron ball will have a slight greater speed than the rubber ball and thus will move a bit further.

4. Probably not as the question mentions the phrase "a smooth floor".

Although it could be said that while this would affect the absolute values of friction, it wouldn't change the relativity, i.e., the fact that the rubber ball will experience more friction that the iron one.

Also, considering it would mean we should probably also consider air resistance. And then the question becomes unsolvable as there'll be unknown variables required then.

For that and the reason above, it should probably not be considered.

10x, @Nathaniel and @Tromik.

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look at the picture so train is moving in one direction with a constant velocity so there will be no force acting on the fall except mg. Now when the train applies the brake and start to descend with a constant negative acceleration $a$ so the acceleration will be in opposite direction to which the train is moving. Since the balls were also moving with the same velocity so due to inertia they will experience an acceleration in the same direction as the train was moving which is $Ma$ and $ma$ so now as you know $F = ma$ so acceleration $a$ of both the ball will be same but the iron ball will move to longer distance as it got more inertia than the rubber ball and you don't need to worry about the air resistance as it not mentioned in the ques and probably it is not in you syllabus too and since it is mentioned that the surface is smooth then you not need to worry about the friction too and if you want an answer with both two condition in action notify me hope it made your doubt clear ans yes your teacher is wright.
You are basically saying $a_{iron} = a_{rubber}$ and $M_{iron} > M_{rubber}$, so therefore $F_{iron} > F_{rubber}$ which is correct. But then you also say that the iron ball will move a longer distance which implies that both the balls will eventually come to a stop while simultaneously claiming to ignore air resistance and friction. So tell me, which force eventually stops the balls (as they'll keep moving at the constant speed otherwise as dictated by one of Newton's Laws of Motion). – YatharthROCK Jul 22 '13 at 14:18