# Why is a train treated as a particle when considering its acceleration?

For the fllowing excercise, I've succesfully answered parts a) and b), but don't understand the answer to part c):

A train travels from Glasgow to Stirling. It starts from rest and accelerates uniformly for the first 9 km of its journey. It then travels for 46.8 km at a uniform velocity, before decelerating uniformly to rest in 7.2 km. The total journey time is 33 minutes.

(a) Sketch a velocity-time graph with appropriate units to represent this journey.

(b) Calculate, in km h-1, the maximum speed reached by the train.

For part c), the answer is that the train is being treated as a particle. I just can't see exactly why this is true. I wondered whether it'd be something related to frictional forces or the conversion of energy but since there's nothing about that in the question I don't believe it is related to them.

• Would "the solutions might be wrong" be an acceptable answer? Aug 11, 2016 at 22:15
• A more creative answer to (c) might be "the track is assumed to run in a perfectly straight line between Glasgow and Stirling" (which is completely wrong, if you look at a map!). Otherwise, a train travelling 46.8 km at uniform velocity would have crashed at the first bend in the track". A more brutal version of the same assumption would be "the person who set the question doesn't know the difference between velocity and speed. Aug 11, 2016 at 22:46

I think what the given answer means, when it assumes the train is a "point particle", is that the length of the train is much less than the length of the track. Otherwise the distance travelled by each point on the train could be significantly less than the distance between Glasgow and Stirling. For example, if the length of the track was 2km and the length of the train was 500m, then each point on the train will travel 1.5km and not 2km.

Without this assumption there is insufficient information to solve the problem. However, it is the kind of assumption which most students make without thinking about it.

The longest passenger trains are about 1.2km in Australia, although around 400m is more typical in Europe. In the present case the journey is 64km in length, so the "point particle" model is realistic.

• Not really very important, but for what it's worth, 400 m is very long for a British passenger train indeed, for example, the Class 377s vary from roughly 80 m to 240 m in length, depending on how many carriages the specific train has. Sorry, as you were Aug 12, 2016 at 23:47

Two things come up in the theory of mechanics when you pass from "particles" to extended objects:

• Rotations Which are handled by considering the torques applied to the system and its moment of inertial.

• Deformations Which are handled with a variety of formalisms, but the starting place is usually the scalar elastic modulli.

The treatment used in the problem ignores the energy put into rotating the wheels (a fairly minor approximation) as well as the possibility of stretching or compressing the train (but, of course, the cars and couplings are engineered to minimize the size of these strains), so it is "treating the train as a particle" in the sense that it is not considering the physics of extended objects.

I can't say I like that as a question, and the commenters on the question have discussed some other issues which are elided in the an attempt to make the question seem relevant to day-to-day life.

• Right. We make models that include properties that are essential to solving the problem, and we ignore the rest. As we ask more detailed questions, we add more properties. In this case, we have a very simple model: the point particle model. Aug 12, 2016 at 0:12

The component of velocity and acceleration tangent to the train track is the same for each car of the train. So, if all you are interested in is the force balance in the tangential direction, it is valid to treat the train as if it were traveling in a straight line. Under these circumstances, treating it as a particle would also be valid.

Actually is not you making the assumption, when answering the question. The author, when he formulated the question, used this assumption. He assumes that you can associate unique values of velocity or acceleration to the train. He does this when he says that "the train" accelerates or "the train" moves with constant velocity. He does not say that a specific point or portion of the train does these but the trains as a whole. This is OK if all the points of the train are moving together, as they do in a rigid body. If there is no rotation involved, the motion of a rigid can be described by taking the motion of the center of mass (or any other point, like the one on the windshield of the engine).