ICE train

German runways test a new ICE train between Munich and Berlin. The train runs to Berlin at a slow speed of 100 km/h. Back from Berlin goes faster. How quickly did the train have to go on a return trip so that the total average train speed for both journeys will be 200 km/h? (The task is from the customer magazine of German Railways ...)

Correct result:

v =  INF km/h

Solution:

v1=100 km/h v2=200 km/h t1=s/v1 t2=s/v v2=2s/(s/v1+s/v) v2=2/(1/v1+1/v) 1/v1+1/v=2/v2 v=1/(2/v2−1/v1)=1/(2/200−1/100)=INF km/hv_{1}=100 \ \text{km/h} \ \\ v_{2}=200 \ \text{km/h} \ \\ t_{1}=s/v_{1} \ \\ t_{2}=s/v \ \\ v_{2}=2s /( s/v_{1} + s/v) \ \\ v_{2}=2 /(1/v_{1} + 1/v) \ \\ 1/v_{1} + 1/v=2/v_{2} \ \\ v=1/ (2/v_{2} - 1/v_{1})=1/ (2/200 - 1/100)=INF \ \text{km/h}

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Dr Math
infinite speed is due that train goes first half at 100 km/h so it cannot achieve average speed 200 km/h, only if second half from Munich take 0 seconds (faster that speed of  light)

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