Degrees of freedom in translational motion

Question

According to this Wikipedia article Degrees of freedom (mechanics):

1. The position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car is defined by the distance along the track.
2. A single particle in space requires three coordinates so it has three degrees of freedom.

According to my Mechanics textbook:

• A rigid body moving in a translational motion has 3 degrees of freedom.

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1. What's the difference between the rigid body in my textbook and the railcar in the Wikipedia article?

2. How many DOFs does a point particle moving in a translational motion have?

• Shouldn't it be treated just like a rigid body in translational motion (since the body doesn't have rotations)?
• If so, how can a constrained point particle have 3 DOFs (the same DOFs as a particle that is not constrained in motion)?

Answer 1

What "degrees of freedom" comes down to is really information. If a train is sitting on a rail initially at position $x=0$ and I tell you

The train moves forward/backward $10 ~\text{meters}$

Then, after only being given one number, you have all the information you need to deduce the final position of the train, namely, $x=\pm 10~\text{meters}.$

On the other hand, if we have a particle in free space initially at position $\mathrm{r}=0$ and I tell you

The particle moves forward/backward $10~\text{meters}$

You don't have enough information to determine the final position of the train. The particle could be in infinitely many places, like $(0,0,10)$ or $(1,2,\sqrt 5)$, etc. One number is now not enough. In order to convey the final position of the particle, you need at least three numbers. This is what three degrees of freedom means.