Projectile — phryzby

solved Projectile

Q. A body is thrown at $v (u₀)$ at $θ (u₀)$ where gravity is $g$ . Where does it land, how high does it go $v_{y} (u₁) = 0$ m/s, and how long is it in the air?

How it's solved

Solved with Projectile

Given

$v_{y} (u₁) = 0$ m/s from “how high does it go”
$θ (u₀)$ , $v (u₀)$ , $g$ yours to vary

Assumed

$x_{x} (u₀) = 0$ m
$x_{y} (u₀) = 0$ m
$x_{y} (d₁) = 0$ m

Find $x_{x} (d₁)$ , $x_{y} (u₁)$ , $Δ t_{f}$ .

Working

parts of a velocity(body, start of up, y) $v_{y} (u₀) = v (u₀) \cdot \sin θ (u₀)$
$v_{y} (u₀) = 20 \cdot \sin 45 °$
$v_{y} (u₀) = 14.1421356$ m/s
falling freely(up, y) $a_{y} (u) = - g$
$a_{y} (u) = - 9.81$
$a_{y} (u) = −9.81$ m/s²
velocity after a time(up, y) $v_{y} (u₁) = v_{y} (u₀) + a_{y} (u) \cdot Δ t_{u}$
$0 = 14.1421356 + (−9.81) \cdot Δ t_{u}$
$Δ t_{u} = 1.44160404$ s
position after a time(up, y) $x_{y} (u₁) = x_{y} (u₀) + v_{y} (u₀) \cdot Δ t_{u} + \frac{a_{y} (u) \cdot {Δ t_{u}}^{2}}{2}$
$x_{y} (u₁) = 0 + 14.1421356 \cdot 1.44160404 + \frac{(−9.81) \cdot {1.44160404}^{2}}{2}$
$x_{y} (u₁) = 10.1936799$ m
falling freely(down, y) $a_{y} (d) = - g$
$a_{y} (d) = - 9.81$
$a_{y} (d) = −9.81$ m/s²
position after a time(down, y) $x_{y} (d₁) = x_{y} (u₁) + v_{y} (u₁) \cdot Δ t_{d} + \frac{a_{y} (d) \cdot {Δ t_{d}}^{2}}{2}$
$0 = 10.1936799 + 0 \cdot Δ t_{d} + \frac{(−9.81) \cdot {Δ t_{d}}^{2}}{2}$
$Δ t_{d} = 1.44160404$ s
parts of a velocity(body, start of up, x) $v_{x} (u₀) = v (u₀) \cdot \cos θ (u₀)$
$v_{x} (u₀) = 20 \cdot \cos 45 °$
$v_{x} (u₀) = 14.1421356$ m/s
falling freely(up, x) $a_{x} (u) = 0$
$a_{x} (u) = 0$ m/s²
position after a time(up, x) $x_{x} (u₁) = x_{x} (u₀) + v_{x} (u₀) \cdot Δ t_{u} + \frac{a_{x} (u) \cdot {Δ t_{u}}^{2}}{2}$
$x_{x} (u₁) = 0 + 14.1421356 \cdot 1.44160404 + \frac{0 \cdot {1.44160404}^{2}}{2}$
$x_{x} (u₁) = 20.3873598$ m
velocity after a time(up, x) $v_{x} (u₁) = v_{x} (u₀) + a_{x} (u) \cdot Δ t_{u}$
$v_{x} (u₁) = 14.1421356 + 0 \cdot 1.44160404$
$v_{x} (u₁) = 14.1421356$ m/s
falling freely(down, x) $a_{x} (d) = 0$
$a_{x} (d) = 0$ m/s²
position after a time(down, x) $x_{x} (d₁) = x_{x} (u₁) + v_{x} (u₁) \cdot Δ t_{d} + \frac{a_{x} (d) \cdot {Δ t_{d}}^{2}}{2}$
$x_{x} (d₁) = 20.3873598 + 14.1421356 \cdot 1.44160404 + \frac{0 \cdot {1.44160404}^{2}}{2}$
$x_{x} (d₁) = 40.7747197$ m
velocity after a time(down, y) $v_{y} (d₁) = v_{y} (u₁) + a_{y} (d) \cdot Δ t_{d}$
$v_{y} (d₁) = 0 + (−9.81) \cdot 1.44160404$
$v_{y} (d₁) = −14.1421356$ m/s
falling freely(flying, y) $a_{y} (f) = - g$
$a_{y} (f) = - 9.81$
$a_{y} (f) = −9.81$ m/s²
velocity after a time(flying, y) $v_{y} (d₁) = v_{y} (u₀) + a_{y} (f) \cdot Δ t_{f}$
$(−14.1421356) = 14.1421356 + (−9.81) \cdot Δ t_{f}$
$Δ t_{f} = 2.88320808$ s

Conclusion

Wherever the throw sent it. Nothing touches it after it leaves the hand, so the angle it left at, the speed it left at, and how hard the world pulls decide the whole flight.

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 :projectile do4  describes "A body is thrown at %{speed} at %{angle} where gravity is %{gravity}. %{range}, %{peak}, and %{time}?"5  given :angle6  given :speed7  given :gravity8  asks range: "Where does it land", peak: "how high does it go", time: "how long is it in the air"9end

solution.rb

1require_relative "../questions/projectile"2require_relative "../science/motion_in_a_plane/projectile"34solve :projectile, using: Projectile do5  objects :body6  axes :x, :y7  body.goes(:up)8  body.goes(:down, after: up)9  body.goes(:flying, made_of: [ up, down ])10  take angle: body.direction(at: flying.start), speed: body.speed(at: flying.start), gravity: g,11       range: body.position(at: flying.end, along: x), peak: body.position(at: up.end, along: y), time: flying.time_taken12  assume body.position(at: flying.start, along: x) => "0 m", body.position(at: flying.start, along: y) => "0 m",13         body.position(at: flying.end, along: y) => "0 m"14  deduce_from "how high does it go", body.velocity(at: up.end, along: y) => "0 m/s"15  find_value :range16  find_value :peak17  find_value :time1819  conclusion "Wherever the throw sent it. Nothing touches it after it leaves the hand, so the angle it left at, the speed it left at, and how hard the world pulls decide the whole flight."20end

playground.rb

1require_relative "../solutions/projectile"2require_relative "../science/motion_in_a_plane/gravity"3require_relative "../pictures/in_view"4require_relative "../pictures/flight"56play_with :projectile do7  title "Projectile"89  input :angle, default: 45.deg, step: 5.deg10  input :speed, 10.0..100.0, in: "m/s", step: 10.0, default: 20.01112  input :world, GRAVITY, default: "earth"13  input :in_view, IN_VIEW1415  input(:gravity) { world }1617  output :range, in: "m"18  output :peak,  in: "m"19  output :time,  in: "s"2021  draw Flight::Picture do22    ground23    flight leaving_at: :angle24    reach "range", :range25    apex "peak", :peak26  end27end

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 body during up along x	$Δ x_{x} (u)$	m	—
position of body at end of up along x	$x_{x} (u₁)$	m	—
position of body at start of up along x	$x_{x} (u₀)$	m	—
displacement of body during up along y	$Δ x_{y} (u)$	m	—
position of body at end of up along y	$x_{y} (u₁)$	m	—
position of body at start of up along y	$x_{y} (u₀)$	m	—
displacement of body during down along x	$Δ x_{x} (d)$	m	—
position of body at end of down along x	$x_{x} (d₁)$	m	—
displacement of body during down along y	$Δ x_{y} (d)$	m	—
position of body at end of down along y	$x_{y} (d₁)$	m	—
displacement of body during flying along x	$Δ x_{x} (f)$	m	—
displacement of body during flying along y	$Δ x_{y} (f)$	m	—
average velocity of body during up along x	$v_{av,x} (u)$	m/s	—
time taken of up	$Δ t_{u}$	s	0 … 1e+05
average velocity of body during up along y	$v_{av,y} (u)$	m/s	—
average velocity of body during down along x	$v_{av,x} (d)$	m/s	—
time taken of down	$Δ t_{d}$	s	0 … 1e+05
average velocity of body during down along y	$v_{av,y} (d)$	m/s	—
average velocity of body during flying along x	$v_{av,x} (f)$	m/s	—
time taken of flying	$Δ t_{f}$	s	0 … 1e+05
average velocity of body during flying along y	$v_{av,y} (f)$	m/s	—
average speed of body during up	$c_{av} (u)$	m/s	—
path length of body during up	$d (u)$	m	—
average speed of body during down	$c_{av} (d)$	m/s	—
path length of body during down	$d (d)$	m	—
average speed of body during flying	$c_{av} (f)$	m/s	—
path length of body during flying	$d (f)$	m	—
change of velocity of body during up along x	$Δ v_{x} (u)$	m/s	—
velocity of body at end of up from ground along x	$v_{x} (u₁)$	m/s	—
velocity of body at start of up from ground along x	$v_{x} (u₀)$	m/s	—
change of velocity of body during up along y	$Δ v_{y} (u)$	m/s	—
velocity of body at end of up from ground along y	$v_{y} (u₁)$	m/s	—
velocity of body at start of up from ground along y	$v_{y} (u₀)$	m/s	—
change of velocity of body during down along x	$Δ v_{x} (d)$	m/s	—
velocity of body at end of down from ground along x	$v_{x} (d₁)$	m/s	—
change of velocity of body during down along y	$Δ v_{y} (d)$	m/s	—
velocity of body at end of down from ground along y	$v_{y} (d₁)$	m/s	—
change of velocity of body during flying along x	$Δ v_{x} (f)$	m/s	—
change of velocity of body during flying along y	$Δ v_{y} (f)$	m/s	—
average acceleration of body during up along x	$a_{av,x} (u)$	m/s²	—
average acceleration of body during up along y	$a_{av,y} (u)$	m/s²	—
average acceleration of body during down along x	$a_{av,x} (d)$	m/s²	—
average acceleration of body during down along y	$a_{av,y} (d)$	m/s²	—
average acceleration of body during flying along x	$a_{av,x} (f)$	m/s²	—
average acceleration of body during flying along y	$a_{av,y} (f)$	m/s²	—
acceleration of body during up along x	$a_{x} (u)$	m/s²	—
acceleration of body during up along y	$a_{y} (u)$	m/s²	—
acceleration of body during down along x	$a_{x} (d)$	m/s²	—
acceleration of body during down along y	$a_{y} (d)$	m/s²	—
acceleration of body during flying along x	$a_{x} (f)$	m/s²	—
acceleration of body during flying along y	$a_{y} (f)$	m/s²	—
acceleration of body during up	$a (u)$	m/s²	—
acceleration of body during flying	$a (f)$	m/s²	—
acceleration of body during down	$a (d)$	m/s²	—
velocity of ground at start of up from ground along x	$v_{g,x} (u₀)$	m/s	—
velocity of ground at start of up from ground	$v_{g} (u₀)$	m/s	—
direction of ground at start of up from ground	$θ_{g} (u₀)$	—	-3.142 … 3.142
velocity of ground at start of up from ground along y	$v_{g,y} (u₀)$	m/s	—
velocity of ground at end of up from ground along x	$v_{g,x} (u₁)$	m/s	—
velocity of ground at end of up from ground	$v_{g} (u₁)$	m/s	—
direction of ground at end of up from ground	$θ_{g} (u₁)$	—	-3.142 … 3.142
velocity of ground at end of up from ground along y	$v_{g,y} (u₁)$	m/s	—
velocity of ground at end of down from ground along x	$v_{g,x} (d₁)$	m/s	—
velocity of ground at end of down from ground	$v_{g} (d₁)$	m/s	—
direction of ground at end of down from ground	$θ_{g} (d₁)$	—	-3.142 … 3.142
velocity of ground at end of down from ground along y	$v_{g,y} (d₁)$	m/s	—
velocity of body at start of up from ground	$v (u₀)$	m/s	—
direction of body at start of up from ground	$θ (u₀)$	—	-3.142 … 3.142
velocity of body at end of up from ground	$v (u₁)$	m/s	—
direction of body at end of up from ground	$θ (u₁)$	—	-3.142 … 3.142
velocity of body at end of down from ground	$v (d₁)$	m/s	—
direction of body at end of down from ground	$θ (d₁)$	—	-3.142 … 3.142
displacement of body during up	$Δ x (u)$	m	—
position of body at end of up	$x (u₁)$	m	—
position of body at start of up	$x (u₀)$	m	—
displacement of body during down	$Δ x (d)$	m	—
position of body at end of down	$x (d₁)$	m	—
displacement of body during flying	$Δ x (f)$	m	—
average velocity of body during up	$v_{av} (u)$	m/s	—
average velocity of body during down	$v_{av} (d)$	m/s	—
average velocity of body during flying	$v_{av} (f)$	m/s	—
change of velocity of body during up	$Δ v (u)$	m/s	—
change of velocity of body during down	$Δ v (d)$	m/s	—
change of velocity of body during flying	$Δ v (f)$	m/s	—
average acceleration of body during up	$a_{av} (u)$	m/s²	—
average acceleration of body during down	$a_{av} (d)$	m/s²	—
average acceleration of body during flying	$a_{av} (f)$	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} (u) = x_{x} (u₁) - x_{x} (u₀)

change of position(up, x)

Δ x_{y} (u) = x_{y} (u₁) - x_{y} (u₀)

change of position(up, y)

Δ x_{x} (d) = x_{x} (d₁) - x_{x} (u₁)

change of position(down, x)

Δ x_{y} (d) = x_{y} (d₁) - x_{y} (u₁)

change of position(down, y)

Δ x_{x} (f) = x_{x} (d₁) - x_{x} (u₀)

change of position(flying, x)

Δ x_{y} (f) = x_{y} (d₁) - x_{y} (u₀)

change of position(flying, y)

v_{av,x} (u) = \frac{Δ x_{x} (u)}{Δ t_{u}}

displacement a time(up, x)

v_{av,y} (u) = \frac{Δ x_{y} (u)}{Δ t_{u}}

displacement a time(up, y)

v_{av,x} (d) = \frac{Δ x_{x} (d)}{Δ t_{d}}

displacement a time(down, x)

v_{av,y} (d) = \frac{Δ x_{y} (d)}{Δ t_{d}}

displacement a time(down, y)

v_{av,x} (f) = \frac{Δ x_{x} (f)}{Δ t_{f}}

displacement a time(flying, x)

v_{av,y} (f) = \frac{Δ x_{y} (f)}{Δ t_{f}}

displacement a time(flying, y)

c_{av} (u) = \frac{d (u)}{Δ t_{u}}

path a time(up, x)

c_{av} (u) = \frac{d (u)}{Δ t_{u}}

path a time(up, y)

c_{av} (d) = \frac{d (d)}{Δ t_{d}}

path a time(down, x)

c_{av} (d) = \frac{d (d)}{Δ t_{d}}

path a time(down, y)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(flying, x)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(flying, y)

Δ v_{x} (u) = v_{x} (u₁) - v_{x} (u₀)

velocity gained(up, x)

Δ v_{y} (u) = v_{y} (u₁) - v_{y} (u₀)

velocity gained(up, y)

Δ v_{x} (d) = v_{x} (d₁) - v_{x} (u₁)

velocity gained(down, x)

Δ v_{y} (d) = v_{y} (d₁) - v_{y} (u₁)

velocity gained(down, y)

Δ v_{x} (f) = v_{x} (d₁) - v_{x} (u₀)

velocity gained(flying, x)

Δ v_{y} (f) = v_{y} (d₁) - v_{y} (u₀)

velocity gained(flying, y)

a_{av,x} (u) = \frac{Δ v_{x} (u)}{Δ t_{u}}

velocity gained a time(up, x)

a_{av,y} (u) = \frac{Δ v_{y} (u)}{Δ t_{u}}

velocity gained a time(up, y)

a_{av,x} (d) = \frac{Δ v_{x} (d)}{Δ t_{d}}

velocity gained a time(down, x)

a_{av,y} (d) = \frac{Δ v_{y} (d)}{Δ t_{d}}

velocity gained a time(down, y)

a_{av,x} (f) = \frac{Δ v_{x} (f)}{Δ t_{f}}

velocity gained a time(flying, x)

a_{av,y} (f) = \frac{Δ v_{y} (f)}{Δ t_{f}}

velocity gained a time(flying, y)

v_{x} (u₁) = v_{x} (u₀) + a_{x} (u) \cdot Δ t_{u}

velocity after a time(up, x)

v_{y} (u₁) = v_{y} (u₀) + a_{y} (u) \cdot Δ t_{u}

velocity after a time(up, y)

v_{x} (d₁) = v_{x} (u₁) + a_{x} (d) \cdot Δ t_{d}

velocity after a time(down, x)

v_{y} (d₁) = v_{y} (u₁) + a_{y} (d) \cdot Δ t_{d}

velocity after a time(down, y)

v_{x} (d₁) = v_{x} (u₀) + a_{x} (f) \cdot Δ t_{f}

velocity after a time(flying, x)

v_{y} (d₁) = v_{y} (u₀) + a_{y} (f) \cdot Δ t_{f}

velocity after a time(flying, y)

x_{x} (u₁) = x_{x} (u₀) + v_{x} (u₀) \cdot Δ t_{u} + \frac{a_{x} (u) \cdot {Δ t_{u}}^{2}}{2}

position after a time(up, x)

x_{y} (u₁) = x_{y} (u₀) + v_{y} (u₀) \cdot Δ t_{u} + \frac{a_{y} (u) \cdot {Δ t_{u}}^{2}}{2}

position after a time(up, y)

x_{x} (d₁) = x_{x} (u₁) + v_{x} (u₁) \cdot Δ t_{d} + \frac{a_{x} (d) \cdot {Δ t_{d}}^{2}}{2}

position after a time(down, x)

x_{y} (d₁) = x_{y} (u₁) + v_{y} (u₁) \cdot Δ t_{d} + \frac{a_{y} (d) \cdot {Δ t_{d}}^{2}}{2}

position after a time(down, y)

x_{x} (d₁) = x_{x} (u₀) + v_{x} (u₀) \cdot Δ t_{f} + \frac{a_{x} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(flying, x)

x_{y} (d₁) = x_{y} (u₀) + v_{y} (u₀) \cdot Δ t_{f} + \frac{a_{y} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(flying, y)

{v_{x} (u₁)}^{2} = {v_{x} (u₀)}^{2} + 2 \cdot a_{x} (u) \cdot (x_{x} (u₁) - x_{x} (u₀))

velocity after a distance(up, x)

{v_{y} (u₁)}^{2} = {v_{y} (u₀)}^{2} + 2 \cdot a_{y} (u) \cdot (x_{y} (u₁) - x_{y} (u₀))

velocity after a distance(up, y)

{v_{x} (d₁)}^{2} = {v_{x} (u₁)}^{2} + 2 \cdot a_{x} (d) \cdot (x_{x} (d₁) - x_{x} (u₁))

velocity after a distance(down, x)

{v_{y} (d₁)}^{2} = {v_{y} (u₁)}^{2} + 2 \cdot a_{y} (d) \cdot (x_{y} (d₁) - x_{y} (u₁))

velocity after a distance(down, y)

{v_{x} (d₁)}^{2} = {v_{x} (u₀)}^{2} + 2 \cdot a_{x} (f) \cdot (x_{x} (d₁) - x_{x} (u₀))

velocity after a distance(flying, x)

{v_{y} (d₁)}^{2} = {v_{y} (u₀)}^{2} + 2 \cdot a_{y} (f) \cdot (x_{y} (d₁) - x_{y} (u₀))

velocity after a distance(flying, y)

x_{x} (u₁) - x_{x} (u₀) = \frac{(v_{x} (u₁) + v_{x} (u₀)) \cdot Δ t_{u}}{2}

mean of the velocities(up, x)

x_{y} (u₁) - x_{y} (u₀) = \frac{(v_{y} (u₁) + v_{y} (u₀)) \cdot Δ t_{u}}{2}

mean of the velocities(up, y)

x_{x} (d₁) - x_{x} (u₁) = \frac{(v_{x} (d₁) + v_{x} (u₁)) \cdot Δ t_{d}}{2}

mean of the velocities(down, x)

x_{y} (d₁) - x_{y} (u₁) = \frac{(v_{y} (d₁) + v_{y} (u₁)) \cdot Δ t_{d}}{2}

mean of the velocities(down, y)

x_{x} (d₁) - x_{x} (u₀) = \frac{(v_{x} (d₁) + v_{x} (u₀)) \cdot Δ t_{f}}{2}

mean of the velocities(flying, x)

x_{y} (d₁) - x_{y} (u₀) = \frac{(v_{y} (d₁) + v_{y} (u₀)) \cdot Δ t_{f}}{2}

mean of the velocities(flying, y)

d (f) = d (u) + d (d)

path in parts(flying)

Δ t_{f} = Δ t_{u} + Δ t_{d}

time in parts(flying)

a (u) = a (f)

steady through its parts(flying, 1)

a (d) = a (f)

steady through its parts(flying, 2)

v_{g,x} (u₀) = v_{g} (u₀) \cdot \cos θ_{g} (u₀)

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

v_{g,y} (u₀) = v_{g} (u₀) \cdot \sin θ_{g} (u₀)

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

v_{g,x} (u₁) = v_{g} (u₁) \cdot \cos θ_{g} (u₁)

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

v_{g,y} (u₁) = v_{g} (u₁) \cdot \sin θ_{g} (u₁)

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

v_{g,x} (d₁) = v_{g} (d₁) \cdot \cos θ_{g} (d₁)

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

v_{g,y} (d₁) = v_{g} (d₁) \cdot \sin θ_{g} (d₁)

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

v_{x} (u₀) = v (u₀) \cdot \cos θ (u₀)

parts of a velocity(body, start of up, x)

v_{y} (u₀) = v (u₀) \cdot \sin θ (u₀)

parts of a velocity(body, start of up, y)

v_{x} (u₁) = v (u₁) \cdot \cos θ (u₁)

parts of a velocity(body, end of up, x)

v_{y} (u₁) = v (u₁) \cdot \sin θ (u₁)

parts of a velocity(body, end of up, y)

v_{x} (d₁) = v (d₁) \cdot \cos θ (d₁)

parts of a velocity(body, end of down, x)

v_{y} (d₁) = v (d₁) \cdot \sin θ (d₁)

parts of a velocity(body, end of down, y)

a_{x} (u) = 0

falling freely(up, x)

a_{y} (u) = - g

falling freely(up, y)

a_{x} (d) = 0

falling freely(down, x)

a_{y} (d) = - g

falling freely(down, y)

a_{x} (f) = 0

falling freely(flying, x)

a_{y} (f) = - g

falling freely(flying, y)

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

magnitude from components(displacement of body during up)

x (u₁) = \sqrt{{x_{x} (u₁)}^{2} + {x_{y} (u₁)}^{2}}

magnitude from components(position of body at end of up)

x (u₀) = \sqrt{{x_{x} (u₀)}^{2} + {x_{y} (u₀)}^{2}}

magnitude from components(position of body at start of up)

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

magnitude from components(displacement of body during down)

x (d₁) = \sqrt{{x_{x} (d₁)}^{2} + {x_{y} (d₁)}^{2}}

magnitude from components(position of body at end of down)

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

magnitude from components(displacement of body during flying)

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

magnitude from components(average velocity of body during up)

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

magnitude from components(average velocity of body during down)

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

magnitude from components(average velocity of body during flying)

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

magnitude from components(change of velocity of body during up)

v (u₁) = \sqrt{{v_{x} (u₁)}^{2} + {v_{y} (u₁)}^{2}}

magnitude from components(velocity of body at end of up from ground)

v (u₀) = \sqrt{{v_{x} (u₀)}^{2} + {v_{y} (u₀)}^{2}}

magnitude from components(velocity of body at start of up from ground)

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

magnitude from components(change of velocity of body during down)

v (d₁) = \sqrt{{v_{x} (d₁)}^{2} + {v_{y} (d₁)}^{2}}

magnitude from components(velocity of body at end of down from ground)

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

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

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

magnitude from components(average acceleration of body during up)

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

magnitude from components(average acceleration of body during down)

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

magnitude from components(average acceleration of body during flying)

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

magnitude from components(acceleration of body during up)

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

magnitude from components(acceleration of body during down)

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

magnitude from components(acceleration of body during flying)

v_{g} (u₀) = \sqrt{{v_{g,x} (u₀)}^{2} + {v_{g,y} (u₀)}^{2}}

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

v_{g} (u₁) = \sqrt{{v_{g,x} (u₁)}^{2} + {v_{g,y} (u₁)}^{2}}

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

v_{g} (d₁) = \sqrt{{v_{g,x} (d₁)}^{2} + {v_{g,y} (d₁)}^{2}}

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