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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

In Inelastic collision cunculate the vilocity
Anshu Reply
explain how a body becomes electrically charged based on the presence of charged particles
Kym Reply
induction
babar
induction
DEMGUE
definitely by induction
Raymond
induction
Raymond
induction
Shah
induction
Korodhso
please why does a needle sinks in water
DEMGUE
induction
Korodhso
induction
Auwal
what are the calculations of Newton's third law of motiow
Murtala Reply
what is dark matter
apex Reply
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
Declan Reply
what is a galaxy
Maduka Reply
what isflow rate of volume
Abcd Reply
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
Mian Reply
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
bimo Reply
is that the answer
nehemiah
why is it proportional
nehemiah Reply
i don't know
Adah
y
nehemiah
what are the relationship between distance and displacement
Usman Reply
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
Thanks a lot
Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
Chidera Reply
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
I have not try that experiment but I think it will magnet....
Rev Reply
Hey Rev. it will
Jeff
I do think so, it will
Chidera
yes it will
lasisi
If a magnet is in a pool of water, would it be able to have a magnetic field?.
Stella Reply
yes Stella it would
Jeff
formula for electric current
Chizzy Reply
what is that about pleace
Fokoua
what are you given?
Kudzy
what is current
Fokoua
I=q/t
saifullahi
Current is the flow of electric charge per unit time.
saifullahi
What are semi conductors
saifullahi
materials that allows charge to flow at varying conditions, temperature for instance.
Mokua
these are materials which have electrical conductivity greater than the insulators but less than metal, in these materials energy band Gap is very narrow as compared to insulators
Sunil
materials that allows charge to flow at varying conditions, temperature for instance.
Obasi
wao so awesome
Fokoua
At what point in the oscillation of beam will a body leave it?
Atambiri
what is gravitational force
Adah
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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