# I The Train/Fly puzzle

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1. Nov 3, 2015

### themagician

I remember back at physics A level we were posed with the question of what happens when a train hits a fly. According to Newton laws the forces are equal, and so, by this theory, the fly should stop the train. We obviously knew that this didn't happen, but it took the whole class waaay too long to understand what was going on, but it was quite fun :)

Anyone got an answer?

2. Nov 3, 2015

### Staff: Mentor

Welcome to the PF.

Please look up "conservation of momentum". Then you can explain it to us!

3. Nov 3, 2015

### rcgldr

The small area of surface on the train that actually collides with the fly moves backwards, stops, and moves forwards again, same as the fly, for a very brief moment.

update - moves backwards from the frame of reference of the train. My second post clarifies this.

Last edited: Nov 4, 2015
4. Nov 3, 2015

### HallsofIvy

Staff Emeritus
Forces are the same, yes, but what happens when you apply the same force to a fly or a train?

5. Nov 3, 2015

### nasu

No, the fly does not need the train by this theory. It is a flawed conclusion from correct premises. :)

6. Nov 3, 2015

### Staff: Mentor

I haven't worked that out, but I am pretty sure that is not the case. Maybe if the fly were made of steel and the train were made of Jello.

7. Nov 3, 2015

### daniellionyang

Consider the free body diagram. Draw it out for each object!

8. Nov 3, 2015

### rcgldr

I meant backwards with respect to the train. It would seem that there is some moment in time where the fly's body and some tiny part of the train are stopped as the fly's molecules change direction. If this is looked from a molecular perspective, then what could be considered to be the overlapped surfaces of "touching" molecules would have to come to a stop, even if the center of the train's molecules didn't come to a stop.

If the perspective is that the fly never actually touches the train, but is repelled by the electrons, then nothing on the train comes to a stop and the fly's velocity is reversed while only slowing down some of the train's molecules.

Last edited: Nov 4, 2015
9. Nov 4, 2015

### Svein

This is an example of inelastic collision. The momentum is conserved, but the speeds after the collision are equal: $m_{train}v_{train}+m_{fly}v_{fly}=(m_{train}+m_{fly})v_{after}$.

10. Nov 4, 2015

### A.T.

That is Newtons 3rd Law only.

No, that is not what Newtons 2nd Law predicts.

11. Nov 4, 2015

### Staff: Mentor

This is only the case in the frame of reference of the moving train, certainly not in the frame of reference of a stationary observer watching the collision from the side of the track.

12. Nov 4, 2015

### rcgldr

Which I clarified in post #8, but I've now added an update to post #3.

One issue is what is meant when an object touches another object. If some part of the fly's body is considered to be in contact with some part of the train during the collision, then there's some moment in time where the point of contact is stopped with respect to the ground. If the situation is viewed from the molecular level, then there is no contact, just a repulsive force due to the electrons and their related fields, and the force from the field changes the direction of the fly's body, without actual contact, only slowing down the trains molecules without stopping them, although I'm not sure what happens to the average position of the affected electrons.

Then again even in a solid, the molecules are bouncing around and I'm wondering if the maximum speed of molecules between collisions exceeds the forward speed of the train, in which case, part of the train is moving backwards for brief moments even without external collisions.

Last edited: Nov 4, 2015
13. Nov 4, 2015

### Staff: Mentor

That part's true.

How does that follow?

14. Nov 4, 2015

### Staff: Mentor

The original question was almost certainly posed assuming that one could neglect what the individual molecules are doing. Within this framework, if the train is considered a rigid body, there is no way the leading edge of the train can ever have zero velocity during the collision.

Certainly the train is more rigid than the fly. So what happens to the fly? Well the fly is deformable, so when its leading edge encounters the leading edge of the train, the velocity at the leading edge of the fly changes instantaneously from its original velocity to the train velocity (assuming that the train has very large mass compared to the fly). But how can this be? The fly would have to have infinite acceleration. Well, no. The explanation is that the amount of mass involved in instantaneously changing the velocity of just the very leading edge of the fly is nil. The remainder of the fly is still traveling at the same velocity it had before its leading edge met the train. What happens is that a compression zone develops starting at the leading edge of the fly and propagating into the fly. This compression zone increases in length at essentially the speed of sound in the fly. The velocity of the part of the fly within the compression zone is equal to that of the train. The velocity of the part of the fly beyond the compression zone is equal to the original velocity of the fly. After the compression zone reaches the trailing edge of the fly, all parts of the fly are traveling with train velocity. During the collision, the force that the train exerts on the fly is equal to the linear density of the fly (assuming that the linear density is uniform) times the speed of sound in the fly times the velocity of the train.

Chet

15. Nov 4, 2015

### rcgldr

Seems that the instant change in velocity would require a rigid train, not one that is just much more rigid than the fly. If the train isn't (infinitely) rigid, then any force exerted on the train results in some deformation at the point of contact. If the difference in velocity is great enough, then a less rigid object can end up penetrating the surface of the more rigid object (leaving a small dent), partly because the less rigid object becomes more dense and rigid due to compression.

Mythbusters had an episode where a piece of straw fired from an air gun at 320 mph penetrated about 1/4 inch into a palm tree. The myth was that the straw could go through the tree, which was busted, but the straw did end up embedded into the tree. Note the record for a high speed train is 374 mph.

http://mythbustersresults.com/episode61

http://kwc.org/mythbusters/2006/09/episode_61_deadly_straw_primar.html

Last edited: Nov 4, 2015
16. Nov 4, 2015

### Staff: Mentor

Actually I was just trying to keep the discussion simple. A rigid train is definitely not required for the same basic principles to apply. In this case, however, the train will also experience a compressive wave that travels backwards from the contact patch, and both the train and the fly will experience an instantaneous change in velocity at the contact patch. The initial velocity at the patch will lie somewhere between that of the train and that of the fly but, of course, much much much closer to the train velocity.

Chet

17. Nov 4, 2015

### jbriggs444

That disparity would presumably fall out from the ratio of mass density times speed of sound in the fly (low) to mass density times speed of sound in the train (high)?

18. Nov 4, 2015

### Staff: Mentor

Yes, that would be part of it. But, in this problem, because of the complexity in geometry (fly shape, contact region that is small compared to train volume available for compressive wave to travel), the problem could not be modelled analytically.

Chet

19. Nov 4, 2015

### rcgldr

I don't get the part about instantaneous change in velocity. Since even the smallest particle, perhaps just one electron, of the fly has some finite mass, then a finite change in vleocity will require a finite amount of time, since the collision force is not infinite. At some point that particle's velocity wrt ground is zero as it changes direction.

The real world effect is that the impact of the fly would cause some tiny abount of deformation on the trains surface during the early phase of the collision. Assuming the deformation is in the elastic range, the surface would recover back to it's original state.

20. Nov 4, 2015

### jbriggs444

If one models the fly and the train as continuous materials that interact through contact forces and which cannot interpenetrate then the conclusion follows.

In the idealized model (with compressible materials) a deformation will occur. That does not mean that the velocity change of an infinitesimal portion of the contacting surfaces is not instantaneous. I believe that this is what Chester has in mind.

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