Vector Addition in Linear Motion: Riverboat Problem

Vector Addition in Linear Motion

Riverboat Problem

Understanding vector addition is essential in analyzing motion when multiple velocities are involved, such as a boat crossing a river with a current. This topic is closely related to Linear Motion and the application of vectors in physics.

• Key Concept: When two velocities act at right angles, the resultant velocity is found using the Pythagorean theorem.

• Given: Boat speed relative to water: 15 mph; River current speed: 15 mph.

• Required: Speed of the boat relative to the riverbank (resultant speed).

Solution Approach

• Represent the boat's velocity and the river's velocity as perpendicular vectors.

• Use the Pythagorean theorem to find the magnitude of the resultant velocity:

$v_{resultant} = \sqrt{v_{boat}^2 + v_{current}^2}$ $v_{resultant} = \sqrt{(15\ \text{mph})^2 + (15\ \text{mph})^2}$ $v_{resultant} = \sqrt{225 + 225}$ $v_{resultant} = \sqrt{450}$ $v_{resultant} \approx 21.2\ \text{mph}$

• Result: The speed of the boat in the river (relative to the riverbank) is approximately 21.2 mph.

Example Application

• This method applies to any situation where two perpendicular velocities combine, such as airplanes flying in wind or swimmers crossing a stream.

Additional info:

• Vector addition is fundamental in physics for analyzing motion in two or more dimensions.

• When vectors are not perpendicular, the law of cosines is used instead.