Talk:Puzzles/Physics puzzles/Train on Circle/Solution

The Hint and the solution, in my opinion, do not answer the question. The question is about average speed and not about average time it takes to travel around a circle.

100km at 100km/h = 1 hour

100km at 200km/h = 1/2 hour

100km at 300km/h = 1/3 hour

Suppose this is a Formula 1 race (the lenght and shape of the circuit are the same in both laps...not like the example given in the answer): A racer completed the first lap with an average speed of 100km (say for example that the lap took him 3 minutes to complete); on the second lap he goes at 300km average(he will take 1 minute and not 3 again, to make the lap); if we do a simple calculation of the average speed and/or time for the 2 lap race: 200km and 2 minutes. Note that only the time is depending on the lenght of the circuit. the average speed is the same even if you choose Sahara as the circuit.

The example given on the answer says that on the begining of the second lap we don't have any more time. Why? There is no time limit. only speed...

A point should be made that the words velocity and speed are NOT interchangable. Velocity is defined as displacement/time and speed as distance/time. In a straight the two will yield the same magnitude answer. In this example the train travels in a circle and therefore the displacement of the train is actually zero.


 * Perhaps a more mathematical answer is in order.


 * If the distance is $$d$$ and the initial speed is $$s_1$$ then $$t_1 = \frac{d}{s_1}$$ which leads to $$t_1 * s_1 = d$$. Now, if the goal is to double the initial speed (the first of two constraints) on average, then


 * $$(s_1 + s_2)/2 = 2*s_1$$
 * $$s_1 + s_2 = 4*s_1$$
 * $$s_2 = 3*s_1$$


 * This is the obvious answer: 300 km/h. But, use the distance equation (the second of two constraints)


 * $$t_2 * s_2 = d$$
 * $$t_2 * s_2 = d$$
 * $$t_2 * 3 * s_1 = d$$
 * $$\frac{d}{s_1} * 3 * s_1 = d$$
 * $$d * 3 = d$$


 * which has no non-trivial solutions (i.e., $$d=0$$). If $$d=0$$ then $$t_2=0$$, which is stated in the solution.


 * But, for the fun of it, let's say you did the first 100 km lap at 100 km/h and did the second 100 km lap at 300 km/h (the "obvious answer"): how long would it take?


 * $$\frac{100 km}{100 km/h} = 1 hr$$
 * $$\frac{100 km}{300 km/h} = 1/3 hr$$


 * So the total time is 1.3333 hours. The average speed is then
 * $$\frac{200 km}{1.3333 hours} = 150 km/hr$$
 * which is clearly contradictory. Cburnett 19:40, 27 Jun 2005 (UTC)

Solution to the puzzle with example
A train goes around a circle, the first time with an average velocity of 100 km/h. How fast should the train be when going around the circle again a second time, so that in total the average velocity for both loops is 200 km/h?

The obvious answer of 300 kms per hour is wrong. The writer is right. The train will have to go infinitely fast to be able to reach average speed of 200 kmph. I am illustrating the solution with an example below.

Lap 1 distance 100 kms avg speed 100 kmph time taken for first lap is 1 hour

Lap 2 distance 100 kms avg speed say 600 kmph Time taken for second lap is 10 minutes.

Total distance travelled in two laps is 200 kms in one hour 10 minutes this the average speed is 200/1hr 10 minutes which is below 200 kmph.

Double the speed in the second lap. See the picture now. Lap 1 distance 100 kms avg speed 100 kmph time taken for first lap is 1 hour

Lap 2 distance 100 kms avg speed say 1200 kmph Time taken for second lap is 5 minutes.

Total distance travelled in two laps is 200 kms in one hour 5 minutes this the average speed is 200/1hr 5 minutes which is below 200 kmph.

Thus if the first lap is travelled in one hour, no matter how much speed you pick up , you cannot reach the average speed of 200 kmph in the circuit.

Anand K. Ghurye moonsign2001@yahoo.com