Across the Current

solved Vectors

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 fast does it actually move, in km/h?

Inputs

boat speed

A

25 km/h

current speed

B

10 km/h

Outputs

speed

R

How it's solved

Solved with Vectors

Given

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

Find $R$ .

Working

$A = 25$ km/h $= 6.94444444$ m/s
$B = 10$ km/h $= 2.77777778$ m/s
law of cosines $R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ$
$R^{2} = {6.94444444}^{2} + {2.77777778}^{2} + 2 \cdot 6.94444444 \cdot 2.77777778 \cdot \cos 120 °$
$R = 21.79$ km/h
$R = 6.05402631$ m/s $= 21.7944947$ km/h
$R \approx 21.8$ km/hto 3 significant figures

Conclusion

The boat's own velocity and the current's add as vectors, head to tail; the current partly opposes the boat, so the sum is less than either would suggest by adding their speeds.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :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. %{speed}, in km/h?"6  given boat_speed: "25 km/h"7  given current_speed: "10 km/h"8  asks speed: "How fast does it actually move"9end

solution.rb

1require_relative "../questions/across_the_current"2require_relative "../science/motion_in_a_plane/vectors"34solve :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 speed: :R8  answer find_value(:speed, in: "km/h", figures: 3)910  conclusion "The boat's own velocity and the current's add as vectors, head to tail; the current " \11             "partly opposes the boat, so the sum is less than either would suggest by adding " \12             "their speeds."13end

playground.rb

1require_relative "../solutions/across_the_current"23play_with :across_the_current do4  title "Across the Current"5end

The laws it drew on

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