University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

ISBN #9780321501219

12th Edition

3,769 Questions

619,728 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This chapter extends the fundamental concepts of motion into multiple dimensions, emphasizing that displacement, velocity, and acceleration are vector quantities with both magnitude and direction. By mastering vector addition and understanding relative motion, students gain the tools needed to analyze scenarios such as curved motion, projectile trajectories, and observer-dependent perspectives. These foundational concepts pave the way for further studies in force and dynamics.

Learning Objectives

Explain how motion extends into two and three dimensions using vector quantities.

Identify and analyze displacement, velocity, and acceleration as vectors with both magnitude and direction.

Apply principles of vector addition to real-world scenarios such as projectile motion and curved paths.

Understand the concept of relative motion from different observers’ perspectives.

Key Concepts

CONCEPT

DEFINITION

Vector Quantity

A quantity that has both magnitude and direction, essential for describing motion in two or three dimensions.

Displacement

The vector measure of the change in position of an object, indicating the straight-line distance and direction from the starting point to the final position.

Velocity

A vector quantity that describes the rate of change of displacement with respect to time, including the direction of motion.

Acceleration

A vector quantity that represents the rate of change of velocity with time, including the direction in which the velocity is changing.

Relative Motion

The concept that motion can appear differently to different observers depending on their frame of reference.

Example Problems

Example 1

A squirrel has $x-$ and $y$ -coordinates $(1.1 \mathrm{m}, 3.4 \mathrm{m})$ at time $t_{1}=0$ and coordinates $(5.3 \mathrm{m},-0.5 \mathrm{m})$ at time $t_{2}=3.0 \mathrm{s} .$ For this time interval, find (a) the components of the average velocity, and $(\mathrm{b})$ the magnitude and direction of the average velocity.

Example 2

A rininoceros is at the origin of coordinates at time $t_{1}=0 .$ For the time interval from $t_{1}=0$ to $t_{2}=12.0 \mathrm{s}$ , the rhino's average velocity has $x-$ component $-3.8 \mathrm{m} / \mathrm{s}$ and $y$ -component 4.9 $\mathrm{m} / \mathrm{s}$ . At time $t_{2}=12.0 \mathrm{s},(\mathrm{a})$ what are the $x$ - and $y$ -coordinates of the rhino? (b) How far is the rhino from the origin?

Example 3

A web page designer creates an animation in which a dot on a computer screen has a position of $\vec{r}=[4.0 \mathrm{cm}+$ $\left(2.5 \mathrm{cm} / \mathrm{s}^{2}\right) t^{2} ] \hat{\imath}+(5.0 \mathrm{cm} / \mathrm{s}) \hat{t}$ . (a) Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0 \mathrm{s} .$ (b) Find the magnitude and direction of the instantaneous velocity at $t=0, t=1.0 \mathrm{s},$ and $t=2.0 \mathrm{s}$ . (c) Sketch the dot's trajectory from $t=0$ to $t=2.0 \mathrm{s}$ , and show the velocities calculated in part $(\mathrm{b})$ .

Example 4

If $\vec{r}=b t^{2} \hat{\imath}+c t^{3} \hat{y},$ where $b$ and $c$ are positive constants, when does the velocity vector make an angle of $45.0^{\circ}$ with the $x-$ and $y$ -axes?

Example 5

A jet plane is flying at a constant altitude. At time $t_{1}=0$ it has components of velocity $v_{x}=90 \mathrm{m} / \mathrm{s}, v_{y}=110 \mathrm{m} / \mathrm{s}$ . At time $t_{2}=30.0 \mathrm{s}$ the components are $v_{x}=-170 \mathrm{m} / \mathrm{s}, v_{y}=40 \mathrm{m} / \mathrm{s}$ . (a) Sketch the velocity vectors at $t_{1}$ and $t_{2}$ . How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

Step-by-Step Explanations

QUESTION

How do you calculate the resultant velocity of an object moving with both horizontal and vertical components?

STEP-BY-STEP ANSWER:

Step 1: Identify the horizontal (vx) and vertical (vy) components of the velocity vector.
Step 2: Use the Pythagorean theorem to calculate the magnitude of the resultant velocity: v = √(vx² + vy²).
Step 3: Determine the direction (angle) of the velocity vector using trigonometry, typically tan⁻¹(vy/vx).
Step 4: Combine the magnitude and direction to express the velocity vector in standard vector form.
Final Answer: The resultant velocity is v = √(vx² + vy²) in magnitude, and its direction is given by the angle tan⁻¹(vy/vx).

Resultant Velocity in Two Dimensions

QUESTION

How do you analyze motion for two different observers in motion relative to each other?

STEP-BY-STEP ANSWER:

Step 1: Define the velocity of each observer relative to a common reference frame.
Step 2: Express the objects' velocity vectors as seen by the observers.
Step 3: Use vector subtraction to find the relative velocity: v_relative = v_object - v_observer.
Step 4: Evaluate both the magnitude and direction of the relative velocity to understand how the motion appears from different frames.
Final Answer: The relative velocity is obtained by subtracting the observer's velocity vector from the object's velocity vector, giving both the magnitude and direction of the motion as seen from the observer's frame.

Relative Motion Analysis

Common Mistakes

Assuming that velocity and speed are the same; neglecting the directional component of vector quantities.
Forgetting to properly resolve vectors into their components when analyzing two-dimensional motion.
Overlooking the importance of relative motion, leading to incorrect assessments of an observer’s frame of reference.
Misapplying the Pythagorean theorem when vectors are not perpendicular.