Relative velocity & optimal-path problems

Fastest Isn’t the Same as Straightest

River-crossing and relative-velocity problems appear almost every JEE cycle, and the time-vs-drift confusion is the single most common mistake students make on them.

A river of width d flows with current speed u. A boat can move at speed v relative to the water, with v > u. In which direction should the boat head to (a) cross in the least possible time, and (b) land exactly at the point directly opposite its start, with zero drift?

The intuitive guess

"To get across as fast as possible, I should angle the boat upstream to counteract the current." Angling upstream only ever costs you time — it can eliminate drift, but it can never be the fastest way across.