4.5 Relative motion in one and two dimensions (Page 3/8)

University physics volume 1 Page 3 / 8

Flying a plane in a wind

A pilot must fly his plane due north to reach his destination. The plane can fly at 300 km/h in still air. A wind is blowing out of the northeast at 90 km/h. (a) What is the speed of the plane relative to the ground? (b) In what direction must the pilot head her plane to fly due north?

Strategy

The pilot must point her plane somewhat east of north to compensate for the wind velocity. We need to construct a vector equation that contains the velocity of the plane with respect to the ground, the velocity of the plane with respect to the air, and the velocity of the air with respect to the ground. Since these last two quantities are known, we can solve for the velocity of the plane with respect to the ground. We can graph the vectors and use this diagram to evaluate the magnitude of the plane’s velocity with respect to the ground. The diagram will also tell us the angle the plane’s velocity makes with north with respect to the air, which is the direction the pilot must head her plane.

Solution

The vector equation is

{\vec{v}}_{PG} = {\vec{v}}_{PA} + {\vec{v}}_{AG},

where P = plane, A = air, and G = ground. From the geometry in [link] , we can solve easily for the magnitude of the velocity of the plane with respect to the ground and the angle of the plane’s heading,

θ .

A compass shows north is up, east to the right, south down, and west to the left. Vectors V sub P G, V sub A G and V sub P A form a triangle. An airplane is shown on vector V sub P G, which points up. V sub P A points up and to the right, at an angle of theta to vector V sub P G. V sub A G points down and left, at an angle of 45 degrees below the horizontal. V sub P G is the vector sum of v sub P A and V sub A G. — Vector diagram for [link] showing the vectors ${\vec{v}}_{PA}, {\vec{v}}_{AG}, and {\vec{v}}_{PG} .$

(a) Known quantities:

| {\vec{v}}_{PA} | = 300 km / h

| {\vec{v}}_{AG} | = 90 km / h

Substituting into the equation of motion, we obtain $| {\vec{v}}_{PG} | = 230 km / h .$

(b) The angle $θ = {tan}^{−1} \frac{63.64}{300} = 12 °$ east of north.

Got questions? Get instant answers now!

Summary

When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified.
Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame.
If S and $S^{'}$ are two reference frames moving relative to each other at a constant velocity, then the velocity of an object relative to S is equal to its velocity relative to $S^{'}$ plus the velocity of $S^{'}$ relative to S.
If two reference frames are moving relative to each other at a constant velocity, then the accelerations of an object as observed in both reference frames are equal.

Key equations

Position vector	$\vec{r} (t) = x (t) \hat{i} + y (t) \hat{j} + z (t) \hat{k}$
Displacement vector	$Δ \vec{r} = \vec{r} (t_{2}) - \vec{r} (t_{1})$
Velocity vector	$\vec{v} (t) = lim_{Δ t \to 0} \frac{\vec{r} (t + Δ t) - \vec{r} (t)}{Δ t} = \frac{d \vec{r}}{d t}$
Velocity in terms of components	$\vec{v} (t) = v_{x} (t) \hat{i} + v_{y} (t) \hat{j} + v_{z} (t) \hat{k}$
Velocity components	$v_{x} (t) = \frac{d x (t)}{d t} v_{y} (t) = \frac{d y (t)}{d t} v_{z} (t) = \frac{d z (t)}{d t}$
Average velocity	${\vec{v}}_{avg} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{1})}{t_{2} - t_{1}}$
Instantaneous acceleration	$\vec{a} (t) = lim_{t \to 0} \frac{\vec{v} (t + Δ t) - \vec{v} (t)}{Δ t} = \frac{d \vec{v} (t)}{d t}$
Instantaneous acceleration, component form	$\vec{a} (t) = \frac{d v_{x} (t)}{d t} \hat{i} + \frac{d v_{y} (t)}{d t} \hat{j} + \frac{d v_{z} (t)}{d t} \hat{k}$
Instantaneous acceleration as second derivatives of position	$\vec{a} (t) = \frac{d^{2} x (t)}{d t^{2}} \hat{i} + \frac{d^{2} y (t)}{d t^{2}} \hat{j} + \frac{d^{2} z (t)}{d t^{2}} \hat{k}$
Time of flight	$T_{tof} = \frac{2 (v_{0} sin θ)}{g}$
Trajectory	$y = (tan θ_{0}) x - [\frac{g}{2 {(v_{0} cos θ_{0})}^{2}}] x^{2}$
Range	$R = \frac{v_{0}^{2} sin 2 θ_{0}}{g}$
Centripetal acceleration	$a_{C} = \frac{v^{2}}{r}$
Position vector, uniform circular motion	$\vec{r} (t) = A cos ω t \hat{i} + A sin ω t \hat{j}$
Velocity vector, uniform circular motion	$\vec{v} (t) = \frac{d \vec{r} (t)}{d t} = − A ω sin ω t \hat{i} + A ω cos ω t \hat{j}$
Acceleration vector, uniform circular motion	$\vec{a} (t) = \frac{d \vec{v} (t)}{d t} = − A ω^{2} cos ω t \hat{i} - A ω^{2} sin ω t \hat{j}$
Tangential acceleration	$a_{T} = \frac{d \| \vec{v} \|}{d t}$
Total acceleration	$\vec{a} = {\vec{a}}_{C} + {\vec{a}}_{T}$
Position vector in frame S is the position vector in frame $S^{'}$ plus the vector from the origin of S to the origin of $S^{'}$	${\vec{r}}_{P S} = {\vec{r}}_{P S^{'}} + {\vec{r}}_{S^{'} S}$
Relative velocity equation connecting two reference frames	${\vec{v}}_{P S} = {\vec{v}}_{P S^{'}} + {\vec{v}}_{S^{'} S}$
Relative velocity equation connecting more than two reference frames	${\vec{v}}_{P C} = {\vec{v}}_{P A} + {\vec{v}}_{A B} + {\vec{v}}_{B C}$
Relative acceleration equation	${\vec{a}}_{P S} = {\vec{a}}_{P S^{'}} + {\vec{a}}_{S^{'} S}$

Questions & Answers

I'm interested in biological psychology and cognitive psychology

Tanya Reply

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

treat them normal, if they want help then give them. that will make everyone happy

Saurabh

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask

	Anthropology Marriage Family Household By Richley Crapo Start Assignment
	6 Physiotherapy Flashcards Set 6 By Rhodes Start Flashcards
	11 Neuroanatomy 11 The Cerebellum By Stephen Voron Start Quiz
©flickr: Dutch	Las Vegas Timeshare By Donyea Sweets Start Test
	Introductory statistics By OpenStax Read Online Course
	16 Dr. Amberg Pharm quiz 2 By Brooke Delaney Start Exam
	16 Sociology 16 Education MCQ By OpenStax Start Quiz
	4 AP Key Terms 04 Tissue Level of Organization By OpenStax Start Key Terms
	2 Microbiology Final 2 By Madison Christian Start Quiz
	Chemistry Final By Briana Hamilton Start Flashcards