Heading across the Current

Q. A motorboat heads north at 25 km/h through a current that flows at 10 km/h toward 60° east of south. How many degrees east of north does it actually move? expects 23.41°

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

Q. A motorboat heads north at 25 km/h $A = 25$ km/h through a current that flows at 10 km/h $B = 10$ km/h toward 60° east of south $φ = 120 °$ . How many degrees east of north does it actually move?

The playground

Inputs

boat speed

A

25 km/h

current speed

B

10 km/h

Outputs

heading

β

The solution

Solved with Vectors

Given

$A = 25$ km/h
$B = 10$ km/h
$φ = 120 °$ from “toward 60° east of south”

Find $β$ .

Working

$A = 25$ km/h $= 6.94444444$ m/s
$B = 10$ km/h $= 2.77777778$ m/s
law of cosines $R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ$
$R^{2} = {6.94444444}^{2} + {2.77777778}^{2} + 2 \cdot 6.94444444 \cdot 2.77777778 \cdot \cos 120 °$
$R = 6.05402631$
law of sines $R \cdot \sin β = B \cdot \sin φ$
$6.05402631 \cdot \sin β = 2.77777778 \cdot \sin 120 °$
$β = 23.41°$

Conclusion

The current pushes the boat eastward as well as slowing it, so its true course turns off north toward the east.

The Ruby

1require_relative "../physics"23question :heading_across_the_current do4  kind :numerical5  describes "A motorboat heads north at %{boat_speed} through a current that flows at %{current_speed} toward 60° east of south. %{heading}?"6  given boat_speed: "25 km/h"7  given current_speed: "10 km/h"8  asks heading: "How many degrees east of north does it actually move"9end

1require_relative "../questions/heading_across_the_current"2require_relative "../science/motion_in_a_plane/vectors"34solve :heading_across_the_current, using: Vectors do5  take boat_speed: :A, current_speed: :B6  deduce_from "toward 60° east of south", phi: "120 deg"7  take heading: :beta8  answer find_value(:heading, in: "deg")910  conclusion "The current pushes the boat eastward as well as slowing it, so its true course " \11             "turns off north toward the east."12end

1require_relative "../solutions/heading_across_the_current"23play_with :heading_across_the_current do4  title "Heading across the Current"5end

The laws

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$mass$	—	—	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$time$	—	—	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$acceleration$	—	—	Vectors
velocity	$velocity$	—	—	Vectors
displacement	$displacement$	—	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$force$	—	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	${average}_{velocity}$	—	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y