Two cars approach a street corner at right angles to each other (Fig. 3–57). Car 1 travels at a speed relative to Earth v₁ₑ = 35 km/h, and car 2 at v₂ₑ = 55 km/h. What is the relative velocity of car 1 as seen by car 2? What is the velocity of car 2 relative to car 1?

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Understand the problem: The goal is to find the relative velocity of car 1 as seen by car 2, and vice versa. This involves vector subtraction of velocities since the cars are moving at right angles to each other.

Define the velocity vectors: Let the velocity of car 1 relative to Earth be \( \vec{v}_{1e} \) and the velocity of car 2 relative to Earth be \( \vec{v}_{2e} \). Since the cars are moving at right angles, we can represent \( \vec{v}_{1e} \) as \( 35 \ \text{km/h} \) along the x-axis and \( \vec{v}_{2e} \) as \( 55 \ \text{km/h} \) along the y-axis.

Calculate the relative velocity of car 1 as seen by car 2: The relative velocity \( \vec{v}_{12} \) is given by \( \vec{v}_{12} = \vec{v}_{1e} - \vec{v}_{2e} \). Subtract the components of \( \vec{v}_{2e} \) from \( \vec{v}_{1e} \): \( \vec{v}_{12} = (35 \ \text{km/h}) \hat{i} - (55 \ \text{km/h}) \hat{j} \).

Calculate the relative velocity of car 2 as seen by car 1: The relative velocity \( \vec{v}_{21} \) is the negative of \( \vec{v}_{12} \), i.e., \( \vec{v}_{21} = -\vec{v}_{12} = -(35 \ \text{km/h}) \hat{i} + (55 \ \text{km/h}) \hat{j} \).

Determine the magnitude and direction of the relative velocities: Use the Pythagorean theorem to find the magnitude of \( \vec{v}_{12} \) or \( \vec{v}_{21} \): \( |\vec{v}_{12}| = \sqrt{(35)^2 + (55)^2} \). To find the direction, calculate the angle \( \theta \) using \( \tan^{-1}(\text{opposite}/\text{adjacent}) \), where opposite and adjacent are the y and x components of the velocity vector.

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주요 개념

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Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. It is calculated by subtracting the velocity vector of the observer from the velocity vector of the object being observed. This concept is crucial for understanding how two objects interact in motion, especially when they are moving at angles to each other.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In the context of relative velocity, it involves adding the velocity vectors of the two cars, taking into account their directions. This is often done using the Pythagorean theorem when the vectors are perpendicular, as in this case.

Coordinate System

A coordinate system provides a framework for describing the position and motion of objects in space. In this problem, a two-dimensional Cartesian coordinate system can be used, where one car's motion is along the x-axis and the other's along the y-axis. Understanding how to represent motion in a coordinate system is essential for accurately calculating relative velocities.