An Umbrella in the Rain

solved Vectors

Q. Rain falls straight down at 35 m/s $A = 35$ m/s, and a wind then blows it sideways $φ = 90 °$ at 12 m/s $B = 12$ m/s. At what angle from the vertical should you tilt your umbrella, in degrees?

Inputs

rain speed

A

35.0 m/s

wind speed

B

12.0 m/s

Outputs

tilt

β

How it's solved

Solved with Vectors

Given

$A = 35$ m/s
$B = 12$ m/s
$φ = 90 °$ from “sideways”

Find $β$ .

Working

law of cosines $R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ$
$R^{2} = 35^{2} + 12^{2} + 2 \cdot 35 \cdot 12 \cdot \cos 90 °$
$R = 37$
law of sines $R \cdot \sin β = B \cdot \sin φ$
$37 \cdot \sin β = 12 \cdot \sin 90 °$
$β = 18.92°$

Conclusion

The rain arrives along the sum of its fall and the wind's push, and the umbrella should face along that sum, tilted into the wind.

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 :umbrella_in_the_rain do4  kind :numerical5  describes "Rain falls straight down at %{rain_speed}, and a wind then blows it sideways at %{wind_speed}. %{tilt}, in degrees?"6  given rain_speed: "35 m/s"7  given wind_speed: "12 m/s"8  asks tilt: "At what angle from the vertical should you tilt your umbrella"9end

solution.rb

1require_relative "../questions/umbrella_in_the_rain"2require_relative "../science/motion_in_a_plane/vectors"34solve :umbrella_in_the_rain, using: Vectors do5  take rain_speed: :A, wind_speed: :B6  deduce_from "sideways", phi: "90 deg"7  take tilt: :beta8  answer find_value(:tilt, in: "deg")910  conclusion "The rain arrives along the sum of its fall and the wind's push, and the umbrella " \11             "should face along that sum, tilted into the wind."12end

playground.rb

1require_relative "../solutions/umbrella_in_the_rain"23play_with :umbrella_in_the_rain do4  title "An Umbrella in the Rain"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