<< Chapter < Page Chapter >> Page >

Problem : A person can swim at a speed “u” in still water. He points across the direction of water stream to cross a river. The water stream flows with a speed “v” in a linear direction. Find the direction in which he actually swims with respect to the direction of stream.

Solution : Let the direction of stream be x-direction and the direction across stream be y-direction. Let us also denote person with "A" and water stream with “B”.

Here,

Speed of the person (A) with respect to stream (B) : v A B = u Speed of stream (B) with respect to ground : v B = v Speed of the person (A) with respect to ground : v C = ?

Using equation, v AB = v A - v B ,

v A = v B + v AB

From the figure,

Relative velocity

tan θ' = v A B v B = u v

The direction in which he actually swims with respect to the direction of stream is

θ = tan - 1 ( u v )

Got questions? Get instant answers now!

Time to cross the river

Problem : A person can swim at a speed 1 m/s in still water. He swims perpendicular to the direction of water stream, flowing at the speed 2 m/s. If the linear distance covered during the motion is 300 m, then find the time taken to cross the river.

Solution : Let the direction of stream be x-direction and the direction across stream be y-direction. Let us also denote ground person with "A" and water stream with “B”. This is clearly the situation corresponding to the least time for crossing the river.

Here,

Speed of the person (A) with respect to stream (B) : v A B = u = 1 m / s Speed of water stream (B) with respect to ground : v B = v = 2 m / s Speed of the person (A) with respect to ground : v A = ?

We note here that the perpendicular linear distance i.e. the width of river is not given. Instead, the linear distance covered during the motion is given. Hence, we need to find the resultant speed in the direction of motion to find time. Using equation for the resultant velocity,

v A = v B + v AB

From the figure, we have :

Relative velocity

v A = { v AB 2 + v B 2 } = { u 2 + v 2 }

v A = { 1 2 + 2 2 } = 5 m / s t = 500 5 = 100 5 s

Got questions? Get instant answers now!

Problem : A person can swim at a speed of √3 m/s in still water. He swims at an angle of 120° from the stream direction while crossing a river. The water stream flows with a speed of 1 m/s. If the river width is 300 m, how long (in seconds) does he take to reach the river bank on the other side ?

Solution : Let the direction of stream be x-direction and the direction across stream be y-direction. Here, we need to know the component of the resultant velocity in the direction perpendicular to the stream.

Relative velocity

This approach, however, would be tedious. We shall use the fact that the component of " v A " in any one of the two mutually perpendicular directions is equal to the sum of the components of v A B and v B in that direction.

v Ay = v A B cos 30 0 v Ay = 3 cos 30 0 = 3 2 m / s

Thus, time taken to cross the river is :

t = Width of the river v Ay t = 300 x 2 3 = 200 s

Got questions? Get instant answers now!

Multiple references

Problem : A boat, capable of sailing at 2 m/s, moves upstream in a river. The water stream flows at 1 m/s. A person walks from the front end to the rear end of the boat at a speed of 1 m/s along the linear direction. What is the speed of the person (m/s) with respect to ground ?

Solution : Let the direction of stream be x-direction and the direction across stream be y-direction. We further denote boat with “A”, stream with “B”, and person with “C”.

We shall work out this problem in two parts. In the first part, we shall find out the velocity of boat (A) with respect to ground and then we shall find out the velocity of person (C) with respect to ground.

Here,

Velocity of boat (A) with respect to stream (B) : v AB = - 2 m / s Velocity of the stream (A) with respect to ground : v A = 1 m / s Velocity of the person (C) with respect to boat (A) : v CA = 1 m / s Velocity of the person (C) with respect to ground : v C = ?

Relative velocity

The velocity of boat with respect to ground is equal to the resultant velocity of the boat as given by :

v A = v A B + v B v A = - 2 + 1 = - 1 m / s

For the motion of person and boat, the velocity of the person with respect to ground is equal to the resultant velocity of (i) velocity of the person (C) with respect to boat (A) and (ii) velocity of the boat (A) with respect to ground. We note here that relative velocity of person with respect to boat is given and that we have already determined the velocity of boat (A) with respect to ground in the earlier step. Hence,

Relative velocity

v C = v C A + v A v C = 1 + ( - 1 ) = 0

Got questions? Get instant answers now!

Minimum time, distance and speed

Problem : A boy swims to reach a point “Q” on the opposite bank, such that line joining initial and final position makes an angle of 45 with the direction perpendicular to the stream of water. If the velocity of water stream is “u”, then find the minimum speed with which the boy should swim to reach his target.

Crossing a river

The boy swims to reach point “Q”.

Solution : Let “A” and “B” denote the boy and the stream respectively. Here, we are required to know the minimum speed of boy, v A B (say “v”) such that he reaches point “Q”. Now, he can adjust his speed with the direction he swims. Let the boy swims at an angle “θ” with a speed “v”.

Crossing a river

The boy swims to reach point “Q”.

Looking at the figure, it can be seen that we can make use of the given angle by taking trigonometric ratio such as tangent, which will involve speed of boy in still water (v) and the speed of water stream (u). This expression may then be used to get an expression for the minimum speed as required.

The slope of resultant velocity, v A , is :

tan 45 0 = v A x v A y = 1

v A x = v A x

Now, the components of velocity in “x” and “y” directions are :

v A x = u - v sin θ

v A y = v cos θ

Putting in the equation we have :

u - v sin θ = v cos θ

Solving for “v”, we have :

v = u sin θ + cos θ

The velocity is minimum for a maximum value of denominator. The denominator is maximum for a particular value of the angle, θ; for which :

đ đ θ sin θ + cos θ = 0

cos θ - sin θ = 0

tan θ = 1

θ = 45 0

It means that the boy swims with minimum speed if he swims in the direction making an angle of 45 with y-direction. His speed with this angle is :

v = u sin 45 0 + cos 45 0 = 2 u 2 = u 2

Got questions? Get instant answers now!

Problem : A boat crosses a river in minimum time, taking 10 minutes during which time the it drifts by 120 m in the direction of stream. On the other hand, boat takes 12.5 minutes while moving across the river. Find (i) width of the river (ii) velocity of boat in still water and (iii) speed of the stream.

Solution : There are three pieces of information about "minimum time", "drift" and "time along shortest path". Individually each of these values translate into three separate equations, which can be solved to find the required values.

The boat takes minimum time, when it sails in the direction perpendicular to the stream (current). The time to cross the river is given by dividing width with component of resultant velocity ( v A y ). The boat, in this case, sails in the perpendicular direction. Hence, the component of resultant velocity is equal to the velocity of boat in still water ( v AB ). The time to cross the river in this case is :

Crossing a river

The swims towards “P”.

t min = d v A B = d v A B = 10

d = 10 v A B

The drift in this time is given by :

x = v B t min

Putting values,

120 = v B x 10

v B = 12 meter / minute

Now we need to use the information on shortest path. It is given that the boat moves across stream in 12.5 minutes. For this boat has to sail upstream at certain angle. The resultant speed is given by :

Crossing a river

The swims towards “P”.

v A = v A B 2 v B 2

and the time taken is :

d v A B 2 v B 2 = 12.5

Substituting for “d” and “ v B ” and squaring on both sides, we have :

v A B 2 12 2 12.5 2 = d 2 = 10 2 v A B 2

v A B 2 12.5 2 10 2 = 12 2 x 12.5 2

v A B = 12 x 12.5 7.5 = 20 meter / minute

and

d = 10 v A B = 10 x 20 = 200 m

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask