Question 1

What is the correct relation for a boat moving in a river?

Accepted Answer

The correct relation is v_boat/ground = v_boat/water + v_water/ground. The boat's velocity relative to water and the stream's velocity relative to ground are added as vectors, producing the velocity actually observed on the shore.

Question 2

Why does aiming perpendicular give the shortest crossing time?

Accepted Answer

A perpendicular aim uses the entire boat speed across the river, so the cross-bank speed is v_b and the crossing time is d/v_b. The stream still carries the boat downstream during this time, so the shortest-time path is not a direct straight path to the opposite point.

Question 3

How can a boat reach a pier directly opposite its starting point?

Accepted Answer

This works only when v_b > v_s. Then sin(theta) = v_s/v_b, with theta measured from the perpendicular toward upstream, and the perpendicular ground speed is sqrt(v_b² - v_s²). If v_b is equal to or less than v_s, a direct crossing is not possible.

Question 4

What happens when the boat speed equals the stream speed?

Accepted Answer

When v_b = v_s, aiming directly upstream cancels the downstream velocity, but the boat then has zero cross-bank velocity and cannot reach the far bank. Direct zero-drift crossing is possible only when v_b is strictly greater than v_s.

Question 5

Why is the boat's path straight when both velocity components are constant?

Accepted Answer

The boat's ground velocity is the constant vector sum of its velocity relative to the water and the stream's velocity relative to the ground. Constant position increments along that resultant produce a straight path parallel to the resultant arrow.

River–Boat Lab: Crossing the Stream, Running Downstream

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