An Umbrella in the Rain

Q. Rain falls straight down at 35 m/s, and a wind then blows it sideways at 12 m/s. At what angle from the vertical should you tilt your umbrella, in degrees? expects 18.92°

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. 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?

The playground

Inputs

rain speed

A

35.0 m/s

wind speed

B

12.0 m/s

Outputs

tilt

β

The solution

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 Ruby

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

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

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