Landing off a Cliff

solved Projectile

Q. A hiker at the edge of a cliff 490 m high $x_{0,y} = 490$ m throws a stone horizontally $v_{0,y} = 0$ m/s at 15 m/s $v_{0,x} = 15$ m/s. Taking g as 9.8 m/s² $g = 9.8$ m/s², how fast is it going when it lands $x_{1,y} = 0$ m, in m/s?

Inputs

speed

v_{0,x}

15.0 m/s

gravity

g

9.8 m/s²

Outputs

landing speed

v_{1}

How it's solved

Solved with Projectile

Given

$v_{0,x} = 15$ m/s
$g = 9.8$ m/s²
$v_{0,y} = 0$ m/s from “horizontally”
$x_{0,y} = 490$ m from “490 m high”
$x_{1,y} = 0$ m from “lands”

Find $v_{1}$ .

Working

falling freely(flying, y) $a_{y} = - g$
$a_{y} = - 9.8$
$a_{y} = −9.8$ m/s²
position after a time(flying, y) $x_{1,y} = x_{0,y} + v_{0,y} \cdot Δ t + \frac{a_{y} \cdot {Δ t}^{2}}{2}$
$0 = 490 + 0 \cdot Δ t + \frac{(−9.8) \cdot {Δ t}^{2}}{2}$
$Δ t = 10$ s
falling freely(flying, x) $a_{x} = 0$
$a_{x} = 0$ m/s²
velocity after a time(flying, x) $v_{1,x} = v_{0,x} + a_{x} \cdot Δ t$
$v_{1,x} = 15 + 0 \cdot 10$
$v_{1,x} = 15$ m/s
velocity after a time(flying, y) $v_{1,y} = v_{0,y} + a_{y} \cdot Δ t$
$v_{1,y} = 0 + (−9.8) \cdot 10$
$v_{1,y} = −98$ m/s
magnitude from components(velocity of stone at end of flying from ground) $v_{1} = \sqrt{{v_{1,x}}^{2} + {v_{1,y}}^{2}}$
$v_{1} = \sqrt{15^{2} + {(−98)}^{2}}$
$v_{1} = 99.1 m/s$
$v_{1} \approx 99.1$ m/sto 3 significant figures

Conclusion

The sideways speed never changes; all the gain is downward, and the speed is the length of the two together.

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 :landing_off_a_cliff do4  kind :numerical5  describes "A hiker at the edge of a cliff 490 m high throws a stone horizontally at %{speed}. Taking g " \6          "as %{gravity}, %{landing_speed}, in m/s?"7  given speed: "15 m/s"8  given gravity: "9.8 m/s2"9  asks landing_speed: "how fast is it going when it lands"10end

solution.rb

1require_relative "../questions/landing_off_a_cliff"2require_relative "../science/motion_in_a_plane/projectile"34solve :landing_off_a_cliff, using: Projectile do5  objects :stone6  axes :x, :y7  stone.goes(:flying)8  take speed: stone.initial_velocity(along: x), gravity: g9  deduce_from "horizontally", stone.initial_velocity(along: y) => "0 m/s"10  deduce_from "490 m high", stone.initial_position(along: y) => "490 m"11  deduce_from "lands", stone.final_position(along: y) => "0 m"12  landing_speed = stone.final_speed13  answer landing_speed:, in: "m/s", figures: 314  conclusion "The sideways speed never changes; all the gain is downward, and the speed is the " \15             "length of the two together."16end

playground.rb

1require_relative "../solutions/landing_off_a_cliff"23play_with :landing_off_a_cliff do4  title "Landing off a Cliff"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	$t$	s	0 … 1e+05	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	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	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	$v_{av}$	m/s	—	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
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
gravity	$g$	m/s²	—	UniformAcceleration
displacement of stone during flying along x	$Δ x_{x}$	m	—
position of stone at end of flying along x	$x_{1,x}$	m	—
position of stone at start of flying along x	$x_{0,x}$	m	—
displacement of stone during flying along y	$Δ x_{y}$	m	—
position of stone at end of flying along y	$x_{1,y}$	m	—
position of stone at start of flying along y	$x_{0,y}$	m	—
average velocity of stone during flying along x	$v_{av,x}$	m/s	—
time taken of flying	$Δ t$	s	0 … 1e+05
average velocity of stone during flying along y	$v_{av,y}$	m/s	—
average speed of stone during flying	$c_{av}$	m/s	—
path length of stone during flying	$d$	m	—
change of velocity of stone during flying along x	$Δ v_{x}$	m/s	—
velocity of stone at end of flying from ground along x	$v_{1,x}$	m/s	—
velocity of stone at start of flying from ground along x	$v_{0,x}$	m/s	—
change of velocity of stone during flying along y	$Δ v_{y}$	m/s	—
velocity of stone at end of flying from ground along y	$v_{1,y}$	m/s	—
velocity of stone at start of flying from ground along y	$v_{0,y}$	m/s	—
average acceleration of stone during flying along x	$a_{av,x}$	m/s²	—
average acceleration of stone during flying along y	$a_{av,y}$	m/s²	—
acceleration of stone during flying along x	$a_{x}$	m/s²	—
acceleration of stone during flying along y	$a_{y}$	m/s²	—
velocity of ground at start of flying from ground along x	$v_{0,g,x}$	m/s	—
velocity of ground at start of flying from ground	$v_{0,g}$	m/s	—
direction of ground at start of flying from ground	$θ_{0,g}$	—	-3.142 … 3.142
velocity of ground at start of flying from ground along y	$v_{0,g,y}$	m/s	—
velocity of ground at end of flying from ground along x	$v_{1,g,x}$	m/s	—
velocity of ground at end of flying from ground	$v_{1,g}$	m/s	—
direction of ground at end of flying from ground	$θ_{1,g}$	—	-3.142 … 3.142
velocity of ground at end of flying from ground along y	$v_{1,g,y}$	m/s	—
velocity of stone at start of flying from ground	$v_{0}$	m/s	—
direction of stone at start of flying from ground	$θ_{0}$	—	-3.142 … 3.142
velocity of stone at end of flying from ground	$v_{1}$	m/s	—
direction of stone at end of flying from ground	$θ_{1}$	—	-3.142 … 3.142
displacement of stone during flying	$Δ x$	m	—
position of stone at end of flying	$x_{1}$	m	—
position of stone at start of flying	$x_{0}$	m	—
average velocity of stone during flying	$v_{av}$	m/s	—
change of velocity of stone during flying	$Δ v$	m/s	—
average acceleration of stone during flying	$a_{av}$	m/s²	—
acceleration of stone during flying	$a$	m/s²	—

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

starting position

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

final position

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

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

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

magnitude from components

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

a_{i} (part 1) = a_{i} (whole)

steady through its parts

v_{o,x} (t) = v_{o} (t) \cdot \cos θ_{o} (t)

parts of a velocity

a_{i,x} (m) = 0

falling freely

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

Δ x_{x} = x_{1,x} - x_{0,x}

change of position(flying, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(flying, y)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(flying, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(flying, y)

c_{av} = \frac{d}{Δ t}

path a time(flying, x)

c_{av} = \frac{d}{Δ t}

path a time(flying, y)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(flying, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(flying, y)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(flying, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(flying, y)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(flying, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(flying, y)

x_{1,x} = x_{0,x} + v_{0,x} \cdot Δ t + \frac{a_{x} \cdot {Δ t}^{2}}{2}

position after a time(flying, x)

x_{1,y} = x_{0,y} + v_{0,y} \cdot Δ t + \frac{a_{y} \cdot {Δ t}^{2}}{2}

position after a time(flying, y)

{v_{1,x}}^{2} = {v_{0,x}}^{2} + 2 \cdot a_{x} \cdot (x_{1,x} - x_{0,x})

velocity after a distance(flying, x)

{v_{1,y}}^{2} = {v_{0,y}}^{2} + 2 \cdot a_{y} \cdot (x_{1,y} - x_{0,y})

velocity after a distance(flying, y)

x_{1,x} - x_{0,x} = \frac{(v_{1,x} + v_{0,x}) \cdot Δ t}{2}

mean of the velocities(flying, x)

x_{1,y} - x_{0,y} = \frac{(v_{1,y} + v_{0,y}) \cdot Δ t}{2}

mean of the velocities(flying, y)

v_{0,g,x} = v_{0,g} \cdot \cos θ_{0,g}

parts of a velocity(ground, start of flying, x)

v_{0,g,y} = v_{0,g} \cdot \sin θ_{0,g}

parts of a velocity(ground, start of flying, y)

v_{1,g,x} = v_{1,g} \cdot \cos θ_{1,g}

parts of a velocity(ground, end of flying, x)

v_{1,g,y} = v_{1,g} \cdot \sin θ_{1,g}

parts of a velocity(ground, end of flying, y)

v_{0,x} = v_{0} \cdot \cos θ_{0}

parts of a velocity(stone, start of flying, x)

v_{0,y} = v_{0} \cdot \sin θ_{0}

parts of a velocity(stone, start of flying, y)

v_{1,x} = v_{1} \cdot \cos θ_{1}

parts of a velocity(stone, end of flying, x)

v_{1,y} = v_{1} \cdot \sin θ_{1}

parts of a velocity(stone, end of flying, y)

a_{x} = 0

falling freely(flying, x)

a_{y} = - g

falling freely(flying, y)

Δ x = \sqrt{{Δ x_{x}}^{2} + {Δ x_{y}}^{2}}

magnitude from components(displacement of stone during flying)

x_{1} = \sqrt{{x_{1,x}}^{2} + {x_{1,y}}^{2}}

magnitude from components(position of stone at end of flying)

x_{0} = \sqrt{{x_{0,x}}^{2} + {x_{0,y}}^{2}}

magnitude from components(position of stone at start of flying)

v_{av} = \sqrt{{v_{av,x}}^{2} + {v_{av,y}}^{2}}

magnitude from components(average velocity of stone during flying)

Δ v = \sqrt{{Δ v_{x}}^{2} + {Δ v_{y}}^{2}}

magnitude from components(change of velocity of stone during flying)

v_{1} = \sqrt{{v_{1,x}}^{2} + {v_{1,y}}^{2}}

magnitude from components(velocity of stone at end of flying from ground)

v_{0} = \sqrt{{v_{0,x}}^{2} + {v_{0,y}}^{2}}

magnitude from components(velocity of stone at start of flying from ground)

a_{av} = \sqrt{{a_{av,x}}^{2} + {a_{av,y}}^{2}}

magnitude from components(average acceleration of stone during flying)

a = \sqrt{{a_{x}}^{2} + {a_{y}}^{2}}

magnitude from components(acceleration of stone during flying)

v_{0,g} = \sqrt{{v_{0,g,x}}^{2} + {v_{0,g,y}}^{2}}

magnitude from components(velocity of ground at start of flying from ground)

v_{1,g} = \sqrt{{v_{1,g,x}}^{2} + {v_{1,g,y}}^{2}}

magnitude from components(velocity of ground at end of flying from ground)