Physics · Motion In A Plane · NEET
You ADD them as vectors. The boat moves relative to the water, and the water moves relative to the ground, so the boat's true velocity over the ground is v(boat,ground) = v(boat,water) + v(water,ground). You only subtract when you want how one object looks from another (that is comparison, not the true ground path). Simple rule: add the current to get the real path; subtract to get the view from another object.
Because you see the rain's velocity relative to yourself, not relative to the ground. Rain velocity as you see it = v(rain) - v(you). If rain falls straight down at 5 m/s and you run forward at 5 m/s, then in your frame the rain has a backward horizontal part of 5 m/s and a downward part of 5 m/s, so it seems to hit you at 45 degrees from the front. This is why you tilt your umbrella forward.
For the shortest path the boat must land directly opposite the start, so its sideways drift must be zero. Point the boat upstream at angle theta from the straight-across direction so that the upstream part cancels the current: v(boat,water) sin(theta) = v(river). So sin(theta) = v(river) / v(boat,water). This only works if v(boat,water) is greater than v(river); otherwise a straight crossing is impossible.
'In water' (v(boat,water)) is how fast the boat moves relative to the water - what the engine or swimmer produces. 'Over ground' (v(boat,ground)) is the actual velocity you would measure from the riverbank, which combines the boat's effort with the current: v(boat,ground) = v(boat,water) + v(water,ground). NEET questions mix these up on purpose, so always label which frame each speed is measured in.
Only the component of the boat's velocity PERPENDICULAR to the banks (across the river) matters for crossing time. Time to cross = width / (perpendicular component). The current, which is parallel to the banks, never changes the crossing time - it only shifts where you land downstream. This is a very common NEET trap.
The speed of a swimmer in still water is 20 m/s. The speed of river water is 10 m/s flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes with respect to north is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is the velocity of one object as seen from another, found by vector subtraction: v(A relative to B) = v(A) - v(B). In a plane you subtract the x-components and y-components separately, then combine them to get magnitude and direction.
The boat's true velocity over ground is v(boat,ground) = v(boat,water) + v(water,ground). Crossing time = river width divided by the across-river component of the boat's velocity, and downstream drift = current speed times crossing time (unless you steer upstream to cancel it).
Find the rain's velocity relative to the person: v(rain,person) = v(rain) - v(person). Point the umbrella along this relative velocity vector. Its angle from vertical is tan(angle) = (your speed) / (rain's downward speed).
The rationalised NCERT chapter reduced this topic, but relative velocity in two dimensions has appeared in real NEET papers (for example the 2019 swimmer question). It is safe to prepare it as a scoring vector application within Motion in a Plane.
When the boat's speed in water is less than the river current speed. Then sin(theta) = v(river)/v(boat,water) would need to be greater than 1, which cannot happen, so the boat can never fully cancel the drift and must always land downstream.
