Any no. of Karts (up to 12 max)may pass finish line simultaneously.

There's no "simultaneously" in the real world, only further/shorter apart. With microcontroller, MOV Rx,Px takes 1 cycle, with 12 carts that's 2 ports - 2 commands - you get resolution of 0.000002s with standard 1MIPS microcontroller. Want faster? Go with the 100MHZ version, you're up to 0.00000002s resolution, but you must probably start taking relativistic effects into account :) If the task is about carts, anything below 0.001s is overkill, if two of them pass the line within the same 0.001s tick (pro sports don't use as much!), you just call it a tie - and this still leaves you 1000 cycles to process data within 1 such tick and tell whether there was 1 or 2 such carts there, to display the score on a LED board etc - even with standard 12MHZ /12clock micro!

Let's think about one thing more...
You get down to 0.000002s resolution with one microcontroller. Say your carts travel very fast, 200km/hour. That's 55.(5)m/s, you get the cart to travel 0.11 mm every tick. How are you going to measure the 0.11mm movement, especially at such speed?