A Loop in the Sky

Q. An aircraft flies a horizontal loop of radius 1.00 km at a steady 900 km/h. Taking g as 9.8 m/s², how many times g is its acceleration? expects 6.38

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Q. An aircraft flies a horizontal loop of radius 1.00 km $r_{c} = 1.00$ km at a steady 900 km/h $v = 900$ km/h. Taking g as 9.8 m/s² $g = 9.8$ m/s², how many times g is its acceleration?

The playground

Inputs

radius

r_{c}

1 km

speed

v

900 km/h

gravity

g

9.8 m/s²

Outputs

times g

q

The solution

Solved with UniformCircularMotion

Given

$r_{c} = 1.00$ km
$v = 900$ km/h
$g = 9.8$ m/s²

Find $q$ .

Working

$r_{c} = 1.00$ km $= 1000$ m
$v = 900$ km/h $= 250$ m/s
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{c}}$
$a_{c} = \frac{250^{2}}{1000}$
$a_{c} = 62.5$ m/s²
measured in gs(round) $q = \frac{a_{c}}{g}$
$q = \frac{62.5}{9.8}$
$q = 6.37755102$
$q \approx 6.38$ to 3 significant figures

Conclusion

A tight turn at speed asks for far more than gravity's pull, which is why pilots feel several times their own weight in one.

The Ruby

1require_relative "../physics"23question :a_loop_in_the_sky do4  kind :numerical5  describes "An aircraft flies a horizontal loop of radius %{radius} at a steady %{speed}. Taking g as " \6          "%{gravity}, %{times_g}?"7  given radius: "1.00 km"8  given speed: "900 km/h"9  given gravity: "9.8 m/s2"10  asks times_g: "how many times g is its acceleration"11end

1require_relative "../questions/a_loop_in_the_sky"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :a_loop_in_the_sky, using: UniformCircularMotion do5  objects :aircraft6  paths :circle7  aircraft.goes(:round, on: circle)8  take radius: circle.radius, speed: aircraft.speed, gravity: g9  times_g = aircraft.against_gravity10  answer times_g:, figures: 311  conclusion "A tight turn at speed asks for far more than gravity's pull, which is why pilots " \12             "feel several times their own weight in one."13end

1require_relative "../solutions/a_loop_in_the_sky"23play_with :a_loop_in_the_sky do4  title "A Loop in the Sky"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	UniformCircularMotion
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
radius	$r$	m	0 … ∞	UniformCircularMotion
speed	$v$	m/s	0 … ∞	UniformCircularMotion
angular speed	$ω$	rad/s	—	UniformCircularMotion
revolutions	$n_{rev}$	—	—	UniformCircularMotion
frequency	$ν$	Hz	0 … ∞	UniformCircularMotion
centripetal acceleration	$a_{c}$	m/s²	—	UniformCircularMotion
against gravity	$q$	—	—	UniformCircularMotion
period	$T$	s	0 … ∞	UniformCircularMotion
gravity	$g$	m/s²	—	UniformCircularMotion
mass	$m$	kg	0 … ∞	UniformCircularMotion
centripetal force	$F_{c}$	N	—	UniformCircularMotion
displacement of aircraft during round	$Δ x$	m	—
position of aircraft at end of round	$x_{1}$	m	—
position of aircraft at start of round	$x_{0}$	m	—
average velocity of aircraft during round	$v_{av}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of aircraft during round	$c_{av}$	m/s	—
path length of aircraft during round	$d$	m	—
speed of aircraft during round	$v$	m/s	0 … ∞
angular speed of aircraft during round	$ω$	rad/s	—
radius of circle	$r_{c}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of aircraft during round	$ν$	Hz	0 … ∞
revolutions of aircraft during round	$n_{rev}$	—	—
centripetal acceleration of aircraft during round	$a_{c}$	m/s²	—
against gravity of aircraft during round	$q$	—	—
centripetal force on aircraft during round	$F_{c}$	N	—
mass of aircraft	$m$	kg	0 … ∞

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

Δ x = x_{1} - x_{0}

change of position(round)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(round)

c_{av} = \frac{d}{Δ t}

path a time(round)

v = ω \cdot r_{c}

speed round the circle(round)

ω = 2 \cdot \frac{π}{T}

once round a period(round)

ν = \frac{1}{T}

rounds a second(round)

ν = \frac{n_{rev}}{Δ t}

rounds over the time(round)

a_{c} = \frac{v^{2}}{r_{c}}

towards the centre(round)

q = \frac{a_{c}}{g}

measured in gs(round)

F_{c} = m \cdot a_{c}

mass times the turning(round)