A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Verified step by step guidance

1. Identify and define the velocity vectors: Let \( \vec{v}_{CE} \) be the velocity of the canoe relative to the Earth, and \( \vec{v}_{RE} \) be the velocity of the river relative to the Earth. Given \( \vec{v}_{CE} = 0.40 \, \text{m/s} \) southeast and \( \vec{v}_{RE} = 0.50 \, \text{m/s} \) east.
2. Break down the vectors into components: Since southeast is 45 degrees south of east, decompose \( \vec{v}_{CE} \) into east and south components using trigonometric functions. The east component is \( 0.40 \cos(45^\circ) \) and the south component is \( 0.40 \sin(45^\circ) \).
3. Set up the equation to find the velocity of the canoe relative to the river, \( \vec{v}_{CR} \), using the relation \( \vec{v}_{CR} = \vec{v}_{CE} - \vec{v}_{RE} \). Substitute the components of \( \vec{v}_{CE} \) and \( \vec{v}_{RE} \) into this equation.
4. Calculate the components of \( \vec{v}_{CR} \) by subtracting the east component of \( \vec{v}_{RE} \) from the east component of \( \vec{v}_{CE} \), and since there is no north-south component for \( \vec{v}_{RE} \), the south component of \( \vec{v}_{CR} \) remains as the south component of \( \vec{v}_{CE} \).
5. Determine the magnitude of \( \vec{v}_{CR} \) using the Pythagorean theorem on the components, and find the direction by calculating the angle of the resultant vector relative to the east using the tangent function.

Key Concepts

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame. In this scenario, we need to determine the canoe's velocity relative to the river, which involves subtracting the river's velocity from the canoe's velocity as observed from the Earth. This concept is crucial for understanding how different frames of reference affect the perceived motion of objects.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this problem, both the canoe's velocity and the river's velocity are vectors with both magnitude and direction. To find the canoe's velocity relative to the river, we must perform vector subtraction, which involves breaking down the velocities into their components and then combining them appropriately.

Components of Motion

Components of motion refer to breaking down a vector into its horizontal and vertical parts, typically along the x (east-west) and y (north-south) axes. For this question, we will resolve the velocities of the canoe and the river into their respective components to facilitate the vector subtraction. Understanding how to work with components is essential for accurately calculating the resultant velocity in two-dimensional motion.