Instants of a Projectile

solved Projectile

Q. A body is thrown at $v (thrown)$ at $θ (thrown)$ where gravity is $g$ . Where does it land, how high does it go $v_{y} (top) = 0$ m/s, and how long is it in the air? And $Δ t_{s}$ after it is thrown, how far out is it, how high is it, how fast is it going across, how fast is it going up, and which way is it heading?

the throw, all of it

Inputs

angle

θ

45.00°

speed

v

20.0 m/s

world

moonmarsearthjupiter

in view

65 m250 m1 km6 km

Outputs

range

x_{x}

40.77 m

peak

x_{y}

10.19 m

time

Δ t_{f}

2.88 s

one moment of it

Inputs

elapsed

Δ t_{s}

1.40 s

Outputs

distance out

x_{x}

19.80 m

height

x_{y}

10.19 m

velocity across

v_{x}

14.1 m/s

velocity up

v_{y}

0.4 m/s

direction

θ

1.65°

How it's solved

Solved with Projectile

Given

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

Assumed

$x_{x} (thrown) = 0$ m
$x_{y} (thrown) = 0$ m
$x_{y} (lands) = 0$ m

Find $x_{x} (lands)$ , $x_{y} (top)$ , $Δ t_{f}$ , $x_{x} (later)$ , $x_{y} (later)$ , $v_{x} (later)$ , $v_{y} (later)$ , $θ (later)$ .

Working

parts of a velocity(body, thrown, y) $v_{y} (thrown) = v (thrown) \cdot \sin θ (thrown)$
$v_{y} (thrown) = 20 \cdot \sin 45 °$
$v_{y} (thrown) = 14.1421356$ m/s
falling freely(flying, y) $a_{y} (f) = - g$
$a_{y} (f) = - 9.81$
$a_{y} (f) = −9.81$ m/s²
position after a time(flying, y) $x_{y} (lands) = x_{y} (thrown) + v_{y} (thrown) \cdot Δ t_{f} + \frac{a_{y} (f) \cdot {Δ t_{f}}^{2}}{2}$
$0 = 0 + 14.1421356 \cdot Δ t_{f} + \frac{(−9.81) \cdot {Δ t_{f}}^{2}}{2}$
$Δ t_{f} = 0, 2.88320808$ sthe start is set aside, as a later time fits
$Δ t_{f} = 2.88320808$ s
parts of a velocity(body, thrown, x) $v_{x} (thrown) = v (thrown) \cdot \cos θ (thrown)$
$v_{x} (thrown) = 20 \cdot \cos 45 °$
$v_{x} (thrown) = 14.1421356$ m/s
falling freely(flying, x) $a_{x} (f) = 0$
$a_{x} (f) = 0$ m/s²
position after a time(flying, x) $x_{x} (lands) = x_{x} (thrown) + v_{x} (thrown) \cdot Δ t_{f} + \frac{a_{x} (f) \cdot {Δ t_{f}}^{2}}{2}$
$x_{x} (lands) = 0 + 14.1421356 \cdot 2.88320808 + \frac{0 \cdot {2.88320808}^{2}}{2}$
$x_{x} (lands) = 40.7747197$ m
falling freely(to top, y) $a_{y} (t) = - g$
$a_{y} (t) = - 9.81$
$a_{y} (t) = −9.81$ m/s²
velocity after a time(to top, y) $v_{y} (top) = v_{y} (thrown) + a_{y} (t) \cdot Δ t_{t}$
$0 = 14.1421356 + (−9.81) \cdot Δ t_{t}$
$Δ t_{t} = 1.44160404$ s
position after a time(to top, y) $x_{y} (top) = x_{y} (thrown) + v_{y} (thrown) \cdot Δ t_{t} + \frac{a_{y} (t) \cdot {Δ t_{t}}^{2}}{2}$
$x_{y} (top) = 0 + 14.1421356 \cdot 1.44160404 + \frac{(−9.81) \cdot {1.44160404}^{2}}{2}$
$x_{y} (top) = 10.1936799$ m
falling freely(so far, x) $a_{x} (s) = 0$
$a_{x} (s) = 0$ m/s²
position after a time(so far, x) $x_{x} (later) = x_{x} (thrown) + v_{x} (thrown) \cdot Δ t_{s} + \frac{a_{x} (s) \cdot {Δ t_{s}}^{2}}{2}$
$x_{x} (later) = 0 + 14.1421356 \cdot 1.4 + \frac{0 \cdot {1.4}^{2}}{2}$
$x_{x} (later) = 19.7989899$ m
falling freely(so far, y) $a_{y} (s) = - g$
$a_{y} (s) = - 9.81$
$a_{y} (s) = −9.81$ m/s²
position after a time(so far, y) $x_{y} (later) = x_{y} (thrown) + v_{y} (thrown) \cdot Δ t_{s} + \frac{a_{y} (s) \cdot {Δ t_{s}}^{2}}{2}$
$x_{y} (later) = 0 + 14.1421356 \cdot 1.4 + \frac{(−9.81) \cdot {1.4}^{2}}{2}$
$x_{y} (later) = 10.1851899$ m
velocity after a time(so far, x) $v_{x} (later) = v_{x} (thrown) + a_{x} (s) \cdot Δ t_{s}$
$v_{x} (later) = 14.1421356 + 0 \cdot 1.4$
$v_{x} (later) = 14.1421356$ m/s
velocity after a time(so far, y) $v_{y} (later) = v_{y} (thrown) + a_{y} (s) \cdot Δ t_{s}$
$v_{y} (later) = 14.1421356 + (−9.81) \cdot 1.4$
$v_{y} (later) = 0.408135624$ m/s
magnitude from components(velocity of body at later from ground) $v (later) = \sqrt{{v_{x} (later)}^{2} + {v_{y} (later)}^{2}}$
$v (later) = \sqrt{{14.1421356}^{2} + {0.408135624}^{2}}$
$v (later) = 14.1480237$ m/s
parts of a velocity(body, later, x) $v_{x} (later) = v (later) \cdot \cos θ (later)$
$14.1421356 = 14.1480237 \cdot \cos θ (later)$
$θ (later) = 1.65°$

Conclusion

Somewhere along the same arc, at a moment you can name. Ask at half the time of flying and the answers are the range halved and the peak exactly — which is where the three the chapter before worked out came from.

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 :motion do4  describes "A body is thrown at %{speed} at %{angle} where gravity is %{gravity}. %{range}, %{peak}, and %{time}? " \5            "And %{elapsed} after it is thrown, %{distance_out}, %{height}, %{velocity_across}, %{velocity_up}, " \6            "and %{direction}?"7  given :angle8  given :speed9  given :gravity10  given :elapsed11  asks range: "Where does it land", peak: "how high does it go", time: "how long is it in the air",12       distance_out: "how far out is it", height: "how high is it",13       velocity_across: "how fast is it going across", velocity_up: "how fast is it going up",14       direction: "which way is it heading"15end

solution.rb

1require_relative "../questions/motion"2require_relative "../science/motion_in_a_plane/projectile"34solve :motion, using: Projectile do5  objects :body6  axes :x, :y7  instants :thrown, :top, :lands8  instants :thrown, :later9  body.goes(:flying, during: thrown..lands)10  body.goes(:to_top, during: thrown..top)11  body.goes(:so_far, during: thrown..later)12  take angle: body.direction(at: thrown), speed: body.speed(at: thrown), gravity: g, elapsed: so_far.time_taken,13       range: body.position(at: lands, along: x), peak: body.position(at: top, along: y), time: flying.time_taken,14       distance_out: body.position(at: later, along: x), height: body.position(at: later, along: y),15       velocity_across: body.velocity(at: later, along: x), velocity_up: body.velocity(at: later, along: y),16       direction: body.direction(at: later)17  assume body.position(at: thrown, along: x) => "0 m", body.position(at: thrown, along: y) => "0 m",18         body.position(at: lands, along: y) => "0 m"19  deduce_from "how high does it go", body.velocity(at: top, along: y) => "0 m/s"20  find_value :range21  find_value :peak22  find_value :time23  find_value :distance_out24  find_value :height25  find_value :velocity_across26  find_value :velocity_up27  find_value :direction2829  conclusion "Somewhere along the same arc, at a moment you can name. Ask at half the time of flying and the answers are the range halved and the peak exactly — which is where the three the chapter before worked out came from."30end

playground.rb

1require_relative "../solutions/motion"2require_relative "../science/motion_in_a_plane/gravity"3require_relative "../pictures/in_view"4require_relative "../pictures/flight"56play_with :motion do7  title "Instants of a Projectile"89  input(:gravity) { world }1011  about "the throw, all of it" do12    input :angle, default: 45.deg, step: 5.deg13    input :speed, 10.0..100.0, in: "m/s", step: 10.0, default: 20.01415    input :world, GRAVITY, default: "earth"16    input :in_view, IN_VIEW1718    output :range, in: "m"19    output :peak,  in: "m"20    output :time,  in: "s"21  end2223  about "one moment of it" do24    input :elapsed, 0.0..20.0, in: "s", step: 0.01, default: 1.4, up_to: :time2526    output :distance_out,    in: "m"27    output :height,          in: "m"28    output :velocity_across, in: "m/s"29    output :velocity_up,     in: "m/s"30    output :direction,       in: "deg"31  end3233  draw Flight::Picture do34    ground35    flight leaving_at: :angle36    now37    reach "range", :range38    apex "peak", :peak39  end40end

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 flying along x	$Δ x_{x} (f)$	m	—
position of body at lands along x	$x_{x} (lands)$	m	—
position of body at thrown along x	$x_{x} (thrown)$	m	—
displacement of body during flying along y	$Δ x_{y} (f)$	m	—
position of body at lands along y	$x_{y} (lands)$	m	—
position of body at thrown along y	$x_{y} (thrown)$	m	—
displacement of body during to top along x	$Δ x_{x} (t)$	m	—
position of body at top along x	$x_{x} (top)$	m	—
displacement of body during to top along y	$Δ x_{y} (t)$	m	—
position of body at top along y	$x_{y} (top)$	m	—
displacement of body during so far along x	$Δ x_{x} (s)$	m	—
position of body at later along x	$x_{x} (later)$	m	—
displacement of body during so far along y	$Δ x_{y} (s)$	m	—
position of body at later along y	$x_{y} (later)$	m	—
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 velocity of body during to top along x	$v_{av,x} (t)$	m/s	—
time taken of to top	$Δ t_{t}$	s	0 … 1e+05
average velocity of body during to top along y	$v_{av,y} (t)$	m/s	—
average velocity of body during so far along x	$v_{av,x} (s)$	m/s	—
time taken of so far	$Δ t_{s}$	s	0 … 1e+05
average velocity of body during so far along y	$v_{av,y} (s)$	m/s	—
average speed of body during flying	$c_{av} (f)$	m/s	—
path length of body during flying	$d (f)$	m	—
average speed of body during to top	$c_{av} (t)$	m/s	—
path length of body during to top	$d (t)$	m	—
average speed of body during so far	$c_{av} (s)$	m/s	—
path length of body during so far	$d (s)$	m	—
change of velocity of body during flying along x	$Δ v_{x} (f)$	m/s	—
velocity of body at lands from ground along x	$v_{x} (lands)$	m/s	—
velocity of body at thrown from ground along x	$v_{x} (thrown)$	m/s	—
change of velocity of body during flying along y	$Δ v_{y} (f)$	m/s	—
velocity of body at lands from ground along y	$v_{y} (lands)$	m/s	—
velocity of body at thrown from ground along y	$v_{y} (thrown)$	m/s	—
change of velocity of body during to top along x	$Δ v_{x} (t)$	m/s	—
velocity of body at top from ground along x	$v_{x} (top)$	m/s	—
change of velocity of body during to top along y	$Δ v_{y} (t)$	m/s	—
velocity of body at top from ground along y	$v_{y} (top)$	m/s	—
change of velocity of body during so far along x	$Δ v_{x} (s)$	m/s	—
velocity of body at later from ground along x	$v_{x} (later)$	m/s	—
change of velocity of body during so far along y	$Δ v_{y} (s)$	m/s	—
velocity of body at later from ground along y	$v_{y} (later)$	m/s	—
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	—
average acceleration of body during to top along x	$a_{av,x} (t)$	m/s²	—
average acceleration of body during to top along y	$a_{av,y} (t)$	m/s²	—
average acceleration of body during so far along x	$a_{av,x} (s)$	m/s²	—
average acceleration of body during so far along y	$a_{av,y} (s)$	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 to top along x	$a_{x} (t)$	m/s²	—
acceleration of body during to top along y	$a_{y} (t)$	m/s²	—
acceleration of body during so far along x	$a_{x} (s)$	m/s²	—
acceleration of body during so far along y	$a_{y} (s)$	m/s²	—
velocity of ground at thrown from ground along x	$v_{g,x} (thrown)$	m/s	—
velocity of ground at thrown from ground	$v_{g} (thrown)$	m/s	—
direction of ground at thrown from ground	$θ_{g} (thrown)$	—	-3.142 … 3.142
velocity of ground at thrown from ground along y	$v_{g,y} (thrown)$	m/s	—
velocity of ground at lands from ground along x	$v_{g,x} (lands)$	m/s	—
velocity of ground at lands from ground	$v_{g} (lands)$	m/s	—
direction of ground at lands from ground	$θ_{g} (lands)$	—	-3.142 … 3.142
velocity of ground at lands from ground along y	$v_{g,y} (lands)$	m/s	—
velocity of ground at top from ground along x	$v_{g,x} (top)$	m/s	—
velocity of ground at top from ground	$v_{g} (top)$	m/s	—
direction of ground at top from ground	$θ_{g} (top)$	—	-3.142 … 3.142
velocity of ground at top from ground along y	$v_{g,y} (top)$	m/s	—
velocity of ground at later from ground along x	$v_{g,x} (later)$	m/s	—
velocity of ground at later from ground	$v_{g} (later)$	m/s	—
direction of ground at later from ground	$θ_{g} (later)$	—	-3.142 … 3.142
velocity of ground at later from ground along y	$v_{g,y} (later)$	m/s	—
velocity of body at thrown from ground	$v (thrown)$	m/s	—
direction of body at thrown from ground	$θ (thrown)$	—	-3.142 … 3.142
velocity of body at lands from ground	$v (lands)$	m/s	—
direction of body at lands from ground	$θ (lands)$	—	-3.142 … 3.142
velocity of body at top from ground	$v (top)$	m/s	—
direction of body at top from ground	$θ (top)$	—	-3.142 … 3.142
velocity of body at later from ground	$v (later)$	m/s	—
direction of body at later from ground	$θ (later)$	—	-3.142 … 3.142
displacement of body during flying	$Δ x (f)$	m	—
position of body at lands	$x (lands)$	m	—
position of body at thrown	$x (thrown)$	m	—
displacement of body during to top	$Δ x (t)$	m	—
position of body at top	$x (top)$	m	—
displacement of body during so far	$Δ x (s)$	m	—
position of body at later	$x (later)$	m	—
average velocity of body during flying	$v_{av} (f)$	m/s	—
average velocity of body during to top	$v_{av} (t)$	m/s	—
average velocity of body during so far	$v_{av} (s)$	m/s	—
change of velocity of body during flying	$Δ v (f)$	m/s	—
change of velocity of body during to top	$Δ v (t)$	m/s	—
change of velocity of body during so far	$Δ v (s)$	m/s	—
average acceleration of body during flying	$a_{av} (f)$	m/s²	—
average acceleration of body during to top	$a_{av} (t)$	m/s²	—
average acceleration of body during so far	$a_{av} (s)$	m/s²	—
acceleration of body during flying	$a (f)$	m/s²	—
acceleration of body during to top	$a (t)$	m/s²	—
acceleration of body during so far	$a (s)$	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} (f) = x_{x} (lands) - x_{x} (thrown)

change of position(flying, x)

Δ x_{y} (f) = x_{y} (lands) - x_{y} (thrown)

change of position(flying, y)

Δ x_{x} (t) = x_{x} (top) - x_{x} (thrown)

change of position(to top, x)

Δ x_{y} (t) = x_{y} (top) - x_{y} (thrown)

change of position(to top, y)

Δ x_{x} (s) = x_{x} (later) - x_{x} (thrown)

change of position(so far, x)

Δ x_{y} (s) = x_{y} (later) - x_{y} (thrown)

change of position(so far, 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)

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

displacement a time(to top, x)

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

displacement a time(to top, y)

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

displacement a time(so far, x)

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

displacement a time(so far, 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)

c_{av} (t) = \frac{d (t)}{Δ t_{t}}

path a time(to top, x)

c_{av} (t) = \frac{d (t)}{Δ t_{t}}

path a time(to top, y)

c_{av} (s) = \frac{d (s)}{Δ t_{s}}

path a time(so far, x)

c_{av} (s) = \frac{d (s)}{Δ t_{s}}

path a time(so far, y)

Δ v_{x} (f) = v_{x} (lands) - v_{x} (thrown)

velocity gained(flying, x)

Δ v_{y} (f) = v_{y} (lands) - v_{y} (thrown)

velocity gained(flying, y)

Δ v_{x} (t) = v_{x} (top) - v_{x} (thrown)

velocity gained(to top, x)

Δ v_{y} (t) = v_{y} (top) - v_{y} (thrown)

velocity gained(to top, y)

Δ v_{x} (s) = v_{x} (later) - v_{x} (thrown)

velocity gained(so far, x)

Δ v_{y} (s) = v_{y} (later) - v_{y} (thrown)

velocity gained(so far, 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)

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

velocity gained a time(to top, x)

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

velocity gained a time(to top, y)

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

velocity gained a time(so far, x)

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

velocity gained a time(so far, y)

v_{x} (lands) = v_{x} (thrown) + a_{x} (f) \cdot Δ t_{f}

velocity after a time(flying, x)

v_{y} (lands) = v_{y} (thrown) + a_{y} (f) \cdot Δ t_{f}

velocity after a time(flying, y)

v_{x} (top) = v_{x} (thrown) + a_{x} (t) \cdot Δ t_{t}

velocity after a time(to top, x)

v_{y} (top) = v_{y} (thrown) + a_{y} (t) \cdot Δ t_{t}

velocity after a time(to top, y)

v_{x} (later) = v_{x} (thrown) + a_{x} (s) \cdot Δ t_{s}

velocity after a time(so far, x)

v_{y} (later) = v_{y} (thrown) + a_{y} (s) \cdot Δ t_{s}

velocity after a time(so far, y)

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

position after a time(flying, x)

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

position after a time(flying, y)

x_{x} (top) = x_{x} (thrown) + v_{x} (thrown) \cdot Δ t_{t} + \frac{a_{x} (t) \cdot {Δ t_{t}}^{2}}{2}

position after a time(to top, x)

x_{y} (top) = x_{y} (thrown) + v_{y} (thrown) \cdot Δ t_{t} + \frac{a_{y} (t) \cdot {Δ t_{t}}^{2}}{2}

position after a time(to top, y)

x_{x} (later) = x_{x} (thrown) + v_{x} (thrown) \cdot Δ t_{s} + \frac{a_{x} (s) \cdot {Δ t_{s}}^{2}}{2}

position after a time(so far, x)

x_{y} (later) = x_{y} (thrown) + v_{y} (thrown) \cdot Δ t_{s} + \frac{a_{y} (s) \cdot {Δ t_{s}}^{2}}{2}

position after a time(so far, y)

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

velocity after a distance(flying, x)

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

velocity after a distance(flying, y)

{v_{x} (top)}^{2} = {v_{x} (thrown)}^{2} + 2 \cdot a_{x} (t) \cdot (x_{x} (top) - x_{x} (thrown))

velocity after a distance(to top, x)

{v_{y} (top)}^{2} = {v_{y} (thrown)}^{2} + 2 \cdot a_{y} (t) \cdot (x_{y} (top) - x_{y} (thrown))

velocity after a distance(to top, y)

{v_{x} (later)}^{2} = {v_{x} (thrown)}^{2} + 2 \cdot a_{x} (s) \cdot (x_{x} (later) - x_{x} (thrown))

velocity after a distance(so far, x)

{v_{y} (later)}^{2} = {v_{y} (thrown)}^{2} + 2 \cdot a_{y} (s) \cdot (x_{y} (later) - x_{y} (thrown))

velocity after a distance(so far, y)

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

mean of the velocities(flying, x)

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

mean of the velocities(flying, y)

x_{x} (top) - x_{x} (thrown) = \frac{(v_{x} (top) + v_{x} (thrown)) \cdot Δ t_{t}}{2}

mean of the velocities(to top, x)

x_{y} (top) - x_{y} (thrown) = \frac{(v_{y} (top) + v_{y} (thrown)) \cdot Δ t_{t}}{2}

mean of the velocities(to top, y)

x_{x} (later) - x_{x} (thrown) = \frac{(v_{x} (later) + v_{x} (thrown)) \cdot Δ t_{s}}{2}

mean of the velocities(so far, x)

x_{y} (later) - x_{y} (thrown) = \frac{(v_{y} (later) + v_{y} (thrown)) \cdot Δ t_{s}}{2}

mean of the velocities(so far, y)

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

parts of a velocity(ground, thrown, x)

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

parts of a velocity(ground, thrown, y)

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

parts of a velocity(ground, lands, x)

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

parts of a velocity(ground, lands, y)

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

parts of a velocity(ground, top, x)

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

parts of a velocity(ground, top, y)

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

parts of a velocity(ground, later, x)

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

parts of a velocity(ground, later, y)

v_{x} (thrown) = v (thrown) \cdot \cos θ (thrown)

parts of a velocity(body, thrown, x)

v_{y} (thrown) = v (thrown) \cdot \sin θ (thrown)

parts of a velocity(body, thrown, y)

v_{x} (lands) = v (lands) \cdot \cos θ (lands)

parts of a velocity(body, lands, x)

v_{y} (lands) = v (lands) \cdot \sin θ (lands)

parts of a velocity(body, lands, y)

v_{x} (top) = v (top) \cdot \cos θ (top)

parts of a velocity(body, top, x)

v_{y} (top) = v (top) \cdot \sin θ (top)

parts of a velocity(body, top, y)

v_{x} (later) = v (later) \cdot \cos θ (later)

parts of a velocity(body, later, x)

v_{y} (later) = v (later) \cdot \sin θ (later)

parts of a velocity(body, later, y)

a_{x} (f) = 0

falling freely(flying, x)

a_{y} (f) = - g

falling freely(flying, y)

a_{x} (t) = 0

falling freely(to top, x)

a_{y} (t) = - g

falling freely(to top, y)

a_{x} (s) = 0

falling freely(so far, x)

a_{y} (s) = - g

falling freely(so far, y)

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

magnitude from components(displacement of body during flying)

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

magnitude from components(position of body at lands)

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

magnitude from components(position of body at thrown)

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

magnitude from components(displacement of body during to top)

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

magnitude from components(position of body at top)

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

magnitude from components(displacement of body during so far)

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

magnitude from components(position of body at later)

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

magnitude from components(average velocity of body during flying)

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

magnitude from components(average velocity of body during to top)

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

magnitude from components(average velocity of body during so far)

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

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

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

magnitude from components(velocity of body at lands from ground)

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

magnitude from components(velocity of body at thrown from ground)

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

magnitude from components(change of velocity of body during to top)

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

magnitude from components(velocity of body at top from ground)

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

magnitude from components(change of velocity of body during so far)

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

magnitude from components(velocity of body at later from ground)

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

magnitude from components(average acceleration of body during flying)

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

magnitude from components(average acceleration of body during to top)

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

magnitude from components(average acceleration of body during so far)

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

magnitude from components(acceleration of body during flying)

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

magnitude from components(acceleration of body during to top)

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

magnitude from components(acceleration of body during so far)

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

magnitude from components(velocity of ground at thrown from ground)

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

magnitude from components(velocity of ground at lands from ground)

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

magnitude from components(velocity of ground at top from ground)

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

magnitude from components(velocity of ground at later from ground)