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Take-home experiment: relative velocity of a boat

Fill a bathtub half-full of water. Take a toy boat or some other object that floats in water. Unplug the drain so water starts to drain. Try pushing the boat from one side of the tub to the other and perpendicular to the flow of water. Which way do you need to push the boat so that it ends up immediately opposite? Compare the directions of the flow of water, heading of the boat, and actual velocity of the boat.

Adding velocities: a boat on a river

A boat is trying to cross a river. Due to the velocity of the river the path traveled by the boat is diagonal. The velocity of the boat, v boat, is equal to zero point seven five meters per second and is in positive y direction. The velocity of the river, v-river, is equal to one point two meters per second and is in positive x direction. The resultant diagonal velocity v total, which makes an angle of theta with the horizontal x axis, is towards north east direction.
A boat attempts to travel straight across a river at a speed 0.75 m/s. The current in the river, however, flows at a speed of 1.20 m/s to the right. What is the total displacement of the boat relative to the shore?

Refer to [link] , which shows a boat trying to go straight across the river. Let us calculate the magnitude and direction of the boat’s velocity relative to an observer on the shore, v tot size 12{v rSub { size 8{"tot"} } } {} . The velocity of the boat, v boat size 12{v rSub { size 8{"boat"} } } {} , is 0.75 m/s in the y size 12{y} {} -direction relative to the river and the velocity of the river, v river size 12{v rSub { size 8{"river"} } } {} , is 1.20 m/s to the right.

Strategy

We start by choosing a coordinate system with its x -axis parallel to the velocity of the river, as shown in [link] . Because the boat is directed straight toward the other shore, its velocity relative to the water is parallel to the y -axis and perpendicular to the velocity of the river. Thus, we can add the two velocities by using the equations v tot = v x 2 + v y 2 size 12{v rSub { size 8{"tot"} } = sqrt {v rSub { size 8{x} } rSup { size 8{2} } +v rSub { size 8{y} } rSup { size 8{2} } } } {} and θ = tan 1 ( v y / v x ) size 12{θ="tan" rSup { size 8{ - 1} } \( v rSub { size 8{y} } /v rSub { size 8{x} } \) } {} directly.

Solution

The magnitude of the total velocity is

v tot = v x 2 + v y 2 , size 12{v rSub { size 8{"tot"} } = sqrt {v rSub { size 8{x} } rSup { size 8{2} } +v rSub { size 8{y} } rSup { size 8{2} } } ","} {}

where

v x = v river = 1 . 20 m/s size 12{v rSub { size 8{x} } =v rSub { size 8{"river"} } =1 "." "20"" m/s"} {}

and

v y = v boat = 0 . 750 m/s. size 12{v rSub { size 8{y} } =v rSub { size 8{ ital "boat"} } =0 "." "750 m/s."} {}

Thus,

v tot = ( 1 . 20 m/s ) 2 + ( 0 . 750 m/s ) 2 size 12{v rSub { size 8{"tot"} } = sqrt { \( 1 "." "20"" m/s" \) rSup { size 8{2} } + \( 0 "." "750"" m/s" \) rSup { size 8{2} } } } {}

yielding

v tot = 1 . 42 m/s. size 12{v rSub { size 8{"tot"} } =1 "." "42"" m/s."} {}

The direction of the total velocity θ size 12{θ} {} is given by:

θ = tan 1 ( v y / v x ) = tan 1 ( 0 . 750 / 1 . 20 ) . size 12{θ="tan" rSup { size 8{ - 1} } \( v rSub { size 8{y} } /v rSub { size 8{x} } \) ="tan" rSup { size 8{ - 1} } \( 0 "." "750"/1 "." "20" \) "."} {}

This equation gives

θ = 32 . . size 12{θ="32" "." 0º} {}

Discussion

Both the magnitude v and the direction θ of the total velocity are consistent with [link] . Note that because the velocity of the river is large compared with the velocity of the boat, it is swept rapidly downstream. This result is evidenced by the small angle (only 32.0º size 12{"32.0º"} {} ) the total velocity has relative to the riverbank.

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Calculating velocity: wind velocity causes an airplane to drift

Calculate the wind velocity for the situation shown in [link] . The plane is known to be moving at 45.0 m/s due north relative to the air mass, while its velocity relative to the ground (its total velocity) is 38.0 m/s in a direction 20 .0º size 12{"20" "." 0 rSup { size 8{o} } } {} west of north.

An airplane is trying to fly north with velocity v p equal to forty five meters per second at angle of one hundred and ten degrees but due to wind velocity v w in south west direction making an angle theta with the horizontal axis it reaches a position in north west direction with resultant velocity v total equal to thirty eight meters per second and the direction is twenty degrees west of north.
An airplane is known to be heading north at 45.0 m/s, though its velocity relative to the ground is 38.0 m/s at an angle west of north. What is the speed and direction of the wind?

Strategy

In this problem, somewhat different from the previous example, we know the total velocity v tot size 12{v rSub { size 8{ bold "tot"} } } {} and that it is the sum of two other velocities, v w size 12{v rSub { size 8{w} } } {} (the wind) and v p size 12{v rSub { size 8{p} } } {} (the plane relative to the air mass). The quantity v p size 12{v rSub { size 8{p} } } {} is known, and we are asked to find v w size 12{v rSub { size 8{w} } } {} . None of the velocities are perpendicular, but it is possible to find their components along a common set of perpendicular axes. If we can find the components of v w size 12{v rSub { size 8{w} } } {} , then we can combine them to solve for its magnitude and direction. As shown in [link] , we choose a coordinate system with its x -axis due east and its y -axis due north (parallel to v p size 12{v rSub { size 8{p} } } {} ). (You may wish to look back at the discussion of the addition of vectors using perpendicular components in Vector Addition and Subtraction: Analytical Methods .)

Questions & Answers

Is earth is an inertial frame?
Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
thanks
Irungu
Most welcome
Sahim
Hey.. I've a question.
Sahim Reply
?
Shii
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
please
Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
I want learn physics
Vinodhini Reply
sir how to understanding clearly
Vinodhini
try to imagine everything you study in 3d
revolutionary
pls give me one title
Vinodhini
displacement acceleration how understand
Vinodhini
vernier caliper usage practically
Vinodhini
karthik sir is there
Vinodhini
what are the solution to all the exercise..?
What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
How to understand physics
Vinodhini Reply
i like physics very much
Vinodhini
i want know physics practically where used in daily life
Vinodhini
I want to teach physics very interesting to studentd
Vinodhini
hy
Aderibigbe
sir
Vinodhini
how can you build interest in physics
Prince
by reading it
Austin
understanding difficult
Vinodhini
vinodhini mam, physics is used in our day to day life in all events..... everything happening around us can be explained in the base of physics..... saying simple stories happening in our daily life and relating it to physics and questioning students about how or why its happening like that can make
revolutionary
your class more interesting
revolutionary
anything send about physics daily life
Vinodhini
How to understand easily
Vinodhini
check out "LMES" youtube channel
revolutionary
even when you see this message in your phone...it works accord to a physics principle. you touch screen works based on physics, your internet works based on physics, etc....... check out google and search for it
revolutionary
hi vinodhini
Karthi
hi
Dada
yea
Solomon
Hi guys
poojaa
hello
Abednego
what is mean by Newtonian principle of Relativity? definition and explanation with example
revolutionary Reply
what is art physics
Akinbulejo Reply
I've been trying to download a good and comprehensive textbook for physics, pls can somebody help me out?
Olanrewaju
try COLLEGE PHYSICS!! I think it will give you an edge.
Lawal
c=1/c1+c2/1+c3 what is the answer
Akinbulejo Reply
got on answers bro
smith
This may seem like a really stupid question, but is mechanical energy the same as potential energy? If not, what is the difference?
Nikki Reply
what is c=1\c1,c=2\c2,c=3\c3
Akinbulejo
mechanical energy is of two types 1: kinetic energy 2: potential energy,so, potential energy is actually the type of mechanical energy ,the mechanical due to position is designated as potential energy
Iram
Thank you!!!!!
Nikki
Can someone possibly walk me through this problem? " A worker drives a 0.500 kg spike into a rail tie with a 2.50 kg sledgehammer. The hammer hits the spike with a speed of 65.0 m/s. If one-third Of the hammer's kinetic energy is converted to the internal energy of rhe hammer and spike.
Nikki
how much does the total internal energy increase
Nikki
you know the mass and the velocity of the hammer. therefore using the equation (mv^2)/2 you can find the kinetic energy. then take one third of this value and that will be your change in internal energy. here, the important thing is that spike is stationary so it does not contribute to initial Energ
Chathu
Thabk you! :)
Nikki
what is the formula for finding the to total capacitance in series arrangement
Austin Reply
Don't know
Ugwu
C = 1/C1+1/C2+1/C3
Nangbun
c=1/c1+1/c2+1/c3
Akinbulejo
what is heat capacity?
smith Reply
Amount of heat that increases the temperature of 1 kg of matter by 1 degree(either celsius or kalvin)
Mateo
it is the ratio between the amount of heat added to an object and the temperature change
Giovanni
what is the different between specific heat capacity and latent heat capacity?
smith
Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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