20. November 2013

Case study: when average speed matters

This case study shows the effects of acceleration, top speed, deceleration on total travel time on short distances at the example of Transrapid in Shanghai. Bottom line: it is the average speed that matters, not the top speed. Groundbreaking result … no! But it is an illustrative example with concrete numbers that shows why reporting top speeds creates illusions.

Everyone knows from school that the average speed of a vehicle decides about the total travel time. It is however interesting to see the effects of this in practice. The Transrapid in Shanghai makes a good example. It accelerates, paces at top speed for a while until it slows down on its way to the destination. The only reasons for using the Transrapid are probably either fun (then top speed might well be the reason to use it) or shortening travel time to Pǔdōng Airport (in which case only average time matters). In the following, let us assume that we only care about travel time. Then the question might be: how much does the top speed effect travel time? This article presents both the maths and a small calculator to give a concrete answer to this question.

In order to answer the question, let us assume that the Transrapid covers the 30 km distance (x) between Long Yang Road and Pudong Airport by first accelerating constantly (with acceleration a) for a certain period (t_acc) until reaching its maximum speed (v_max), then travelling at maximum speed as long as possible (t_top) until it decelerates constantly (with deceleration d) for a certain period (t_dec) such that it stops exactly at the Airport. We distinguish between the top speed (v_top) due to technical or regulatory limitations and the maximum speed reached (v_max). The latter is bounded as the Transrapid needs to be able to have enough track left after the acceleration period in order to come to a halt. The following diagram illustrates the model:

Now, given the acceleration, top speed, and deceleration of the Transrapid, we are able to compute the time accelerating, the time at top speed (if zero then the Transrapid never reaches top speed), and the total travel time, which is just the sum of these times.

In the following, the acceleration is assumed to be equal to the deceleration. The acceleration itself has been derived from an info the transrapid.de website from ThyssenKrupp. They state that the Transrapid reaches 300 km/h after a distance of 5km — this corresponds to an acceleration of 9000 km/h/h).

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ThyssenKrupp also correctly states that fast acceleration can be a large advantage of the Transrapid. However, in the following, we will see how fast we can get with the reported acceleration. The top speed has been taken from the train operator’s home page to be 430 km/h. Good old Newton provides the desired answer:

x     =    30 km      (distance)
a     =  9000 km/h/h  (acceleration)
v_max =   430 km/h    (top speed)
d     =  9000 km/h/h  (deceleration)
-------------------------------------
v_max =   430 km/h    (maximum speed)
t_acc = 00h 02m 52s   (accelerating)
t_top = 00h 01m 19s   (at top speed)
t_dec = 00h 02m 52s   (decelerating)
-------------------------------------
T     = 00h 07m 03s   (travel time)

However, at certain times of the day, the top speed Shanghai Transrapid only operates at a top speed of 300 km/h (see the train operator’s home page). How does this limit affect the travel times?

x     =    30 km      (distance)
a     =  9000 km/h/h  (acceleration)
v_max =   300 km/h    (top speed)
d     =  9000 km/h/h  (deceleration)
-------------------------------------
v_max =   300 km/h    (maximum speed)
t_acc = 00h 02m 00s   (accelerating)
t_top = 00h 04m 00s   (at top speed)
t_dec = 00h 02m 00s   (decelerating)
-------------------------------------
T     = 00h 08m 00s   (travel time)

The travel times specified on Wikipedia report that the maglev train takes 7:20 minutes when allowed to reach a top speed of 430 km/h, and takes 8:10 minutes when limited to 300 km/h. If the Wikipedia numbers can be trusted, then this is a good indication that the model presented here and the assumptions about acceleration and deceleration are sound.

The interesting fact is that one can only save less than a minute despite of a more than 40% increase in the top speed. Given the time that is usually wasted at the airport waiting, there is not much reason to speed up traveling any further towards Pudong Airport. And what is that minute worth in the other direction?

Now, let us go a little bit more in the extreme. Assume that we have a newer Transrapid that manages to go at 600 km/h, but without improving acceleration and deceleration.

x     =    30 km      (distance)
a     =  9000 km/h/h  (acceleration)
v_max =   600 km/h    (top speed)
d     =  9000 km/h/h  (deceleration)
-------------------------------------
v_max =   519 km/h    (maximum speed)
t_acc = 00h 03m 27s   (accelerating)
t_top = 00h 00m 00s   (at top speed)
t_dec = 00h 03m 27s   (decelerating)
-------------------------------------
T     = 00h 06m 55s   (travel time)

The model predicts a saving of only eight seconds as compared to a Transrapid with top speed at 430km/h. It furthermore predicts that the Transrapid is only able to reach 519 km/h with the given acceleration/deceleration before it has to start slowing down. In order to reach a speed of 600 km/h, the train would need to be able to accelerate from 0-100 km/h in only 30 seconds:

x     =    30 km      (distance)
a     = 12000 km/h/h  (acceleration)
v_max =   600 km/h    (top speed)
d     = 12000 km/h/h  (deceleration)
-------------------------------------
v_max =   600 km/h    (maximum speed)
t_acc = 00h 03m 00s   (accelerating)
t_top = 00h 00m 00s   (at top speed)
t_dec = 00h 03m 00s   (decelerating)
-------------------------------------
T     = 00h 06m 00s   (travel time)

As a conclusion, whether the Shanghai Transrapid now reaches 430 km/h or 300 km/h does not make a large difference in the travel time. This is a known fact, but this article played a little with the numbers. You can repeat the experiments with a Python script.

As to the maths, there are two cases. In the first case, the Transrapid never actually reaches its top speed. Then the total time T is:

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In the second case, the Transrapid reaches its top speed. Then the total time T is:

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Use at your own risk.

Update: I hacked together a small online calculator that computes the travel times as described above. You can play with different acceleration, deceleration, top speed, and with different track lengths. For the sake of simplicity, only integer numbers are reported.