Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Problem 73a

Chapter 3, Problem 73a

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 3–46). With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the car roofs and the horizontal distance he must clear is 22 m.

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Textbook Question

A person in the passenger basket of a hot-air balloon throws a ball horizontally outward from the basket with speed 12.0 m/s (Fig. 3–64). What initial velocity (magnitude and direction) does the ball have relative to a person standing on the ground if the hot-air balloon is descending at 3.0 m/s relative to the ground?

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Textbook Question

At serve, a tennis player aims to hit the ball horizontally. What minimum speed is required for the ball to clear the 0.90-m-high net about 15.0 m from the server if the ball is 'launched' from a height of 2.30 m? Where will the ball land if it just clears the net (and will it be 'good' in the sense that it lands within 7.0 m of the net)? How long will it be in the air? See Fig. 3–50.

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Textbook Question

A motorboat whose speed in still water is 4.30 m/s must aim upstream at an angle of 23.5° (with respect to a line perpendicular to the shore) in order to travel directly across the stream. What is the resultant speed of the boat with respect to the shore? (See Fig. 3–33.)

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