How Does Train Length Affect Speed on a Sine-Shaped Rollercoaster Track?

[Mantaray]

Homework Statement

Imagine there is a piece of sine-shaped rollercoaster track. We compare trains of different length which all have the same average speed riding over the track. If all friction is neglected, is the difference between the maximum and minimum speed:

A. bigger for longer trains
B. smaller for longer trains
C. does not depend on the length of the train
D. depends on the length of the train but not as stated in A or B.

Homework Equations

Not sure..

The Attempt at a Solution

My guess was that longer trains would have their wagons distributed more evenly on the track, such that the bumps would have less effect on the average speed. However, the answer is D and I do not understand why?

 

[tiny-tim]

Hi Mantaray!

My guess was that longer trains would have their wagons distributed more evenly on the track, such that the bumps would have less effect on the average speed. However, the answer is D and I do not understand why?

Hint: what happens if the train is a whole wave-length long?

(btw, your answer really ought to specify some principle of physics, with a little bit of application)

 

[Mantaray]

Thanks! You're right I really should have applied some principles, but I had no clue which one(s) to apply here..

If the train is a whole wave-length long both the minimum and maximum speed will be equal to the average speed because there is no resultant force acting on the train at all times.

Because if the track would look like \/\/\/\/, one \/ would be one wavelength. And as the train is one wave-length long, it'd cover half a \ (let's say this is downhill) and the other half of the train would have to cover half a / (uphill). Which explains why there is no net force.

This is very much like resonance, but I am not sure how to explain the previous paragraph using proper physics arguments.

 

[tiny-tim]

Thanks! You're right I really should have applied some principles, but I had no clue which one(s) to apply here..

Now why did I suspect that?

Use conservation of energy ​

 

[Mantaray]

Let's say that the train covers a whole wave-length of track. This means that the height of the center of gravity will not shift vertically, which means that the amount of potential energy remains constant. There is no friction, so the amount of kinetic energy stays constant too, which means that the speed remains the same. But only when the train is a whole wave-length long or a multiple thereof.

This should be complete right?

 

[tiny-tim]

Let's say that the train covers a whole wave-length of track. This means that the height of the center of gravity will not shift vertically, which means that the amount of potential energy remains constant. There is no friction, so the amount of kinetic energy stays constant too, which means that the speed remains the same. But only when the train is a whole wave-length long or a multiple thereof.

That's right!

And then you can use the maximim and minimum heights of the centre of mass for a train of any other length.