Final answer:
The velocity of the current in a river defines the speed and direction of the river's flow. When a boat tries to cross the river, its total velocity is the vector sum of its own speed and the river's current, often resulting in a diagonal rather than straight-line path.
Step-by-step explanation:
The velocity of the current in a river is a vector quantity that represents both the speed of the water's flow and its direction relative to the riverbank. When a boat attempts to cross a river directly from one bank to the other, its total velocity is the combination of its velocity relative to the water and the water's velocity relative to the riverbank. For instance, if a boat tries to travel straight across a river at a speed of 0.75 m/s and the river current flows at a speed of 1.20 m/s to the right, the boat will be affected by the current and will end up moving diagonally rather than in a straight path across the river.
Using vector addition, the total velocity of the boat can be calculated. Assuming the boat is moving perpendicular to the current, the resulting total velocity will have a magnitude determined by the Pythagorean theorem, and its direction can be found using trigonometric functions such as arctan to determine the angle relative to the riverbank.
Example Calculation
Suppose the boat's velocity is represented by Vboat = 0.75 m/s and the river's velocity is Vriver = 1.20 m/s. The angle θ with which the boat's total velocity makes with respect to the riverbank can be found using the arctan of the ratio of Vboat and Vriver:
θ = arctan(Vboat/Vriver)
The actual calculation would yield an angle of less than 45° since the river's current is faster than the boat's speed perpendicular to it, indicating that the boat is swept downstream even as it tries to cross.