As High as It Goes Far

solved Projectile

Q. A cricketer can throw a ball at most $θ (f₀) = 45 °$ 100 m $x_{x} (f₁) = 100$ m. How high $θ (h₀) = 90 °$ , $v_{y} (h₁) = 0$ m/s can they throw the same ball $v (h₀) = v (f₀)$ , in metres?

Inputs

farthest

x_{x}

100.00 m

Outputs

height

x_{y}

How it's solved

Solved with Projectile

Given

$x_{x} (f₁) = 100$ m
$θ (f₀) = 45 °$ from “at most”
$θ (h₀) = 90 °$ from “How high”
$v_{y} (h₁) = 0$ m/s from “How high”
$v (h₀) = v (f₀)$ from “the same ball”

Assumed

$g = 9.8$ m/s²
$x_{x} (f₀) = 0$ m
$x_{y} (f₀) = 0$ m
$x_{y} (f₁) = 0$ m
$x_{y} (h₀) = 0$ m

Find $x_{y} (h₁)$ .

Working

falling freely(far, y) $a_{y} (f) = - g$
$a_{y} (f) = - 9.8$
$a_{y} (f) = −9.8$ m/s²
falling freely(far, x) $a_{x} (f) = 0$
$a_{x} (f) = 0$ m/s²
parts of a velocity(ball, start of far, y) $v_{y} (f₀) = v (f₀) \cdot \sin θ (f₀)$
position after a time(far, y) $x_{y} (f₁) = x_{y} (f₀) + v_{y} (f₀) \cdot Δ t_{f} + \frac{a_{y} (f) \cdot {Δ t_{f}}^{2}}{2}$
position after a time(far, x) $x_{x} (f₁) = x_{x} (f₀) + v_{x} (f₀) \cdot Δ t_{f} + \frac{a_{x} (f) \cdot {Δ t_{f}}^{2}}{2}$
parts of a velocity(ball, start of far, x) $v_{x} (f₀) = v (f₀) \cdot \cos θ (f₀)$
$v (f₀) = 31.3049517$ m/s
the same ball $v (h₀) = v (f₀)$
$v (h₀) = 31.3049517$
$v (h₀) = 31.3049517$ m/s
parts of a velocity(ball, start of high, y) $v_{y} (h₀) = v (h₀) \cdot \sin θ (h₀)$
$v_{y} (h₀) = 31.3049517 \cdot \sin 90 °$
$v_{y} (h₀) = 31.3049517$ m/s
falling freely(high, y) $a_{y} (h) = - g$
$a_{y} (h) = - 9.8$
$a_{y} (h) = −9.8$ m/s²
velocity after a time(high, y) $v_{y} (h₁) = v_{y} (h₀) + a_{y} (h) \cdot Δ t_{h}$
$0 = 31.3049517 + (−9.8) \cdot Δ t_{h}$
$Δ t_{h} = 3.19438282$ s
position after a time(high, y) $x_{y} (h₁) = x_{y} (h₀) + v_{y} (h₀) \cdot Δ t_{h} + \frac{a_{y} (h) \cdot {Δ t_{h}}^{2}}{2}$
$x_{y} (h₁) = 0 + 31.3049517 \cdot 3.19438282 + \frac{(−9.8) \cdot {3.19438282}^{2}}{2}$
$x_{y} (h₁) = 50.00 m$

Conclusion

The farthest throw, at forty-five degrees, and the highest, straight up, both depend on the square of the speed over the pull, and the height is always half the range.

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 :as_high_as_it_goes_far do4  kind :numerical5  describes "A cricketer can throw a ball at most %{farthest}. %{height}, in " \6          "metres?"7  given farthest: "100 m"8  asks height: "How high can they throw the same ball"9end

solution.rb

1require_relative "../questions/as_high_as_it_goes_far"2require_relative "../science/motion_in_a_plane/projectile"34solve :as_high_as_it_goes_far, using: Projectile do5  objects :ball6  axes :x, :y7  ball.goes(:far)8  ball.goes(:high)9  take farthest: ball.position(at: far.end, along: x)10  assume g: "9.8 m/s2"11  assume ball.position(at: far.start, along: x) => "0 m", ball.position(at: far.start, along: y) => "0 m",12         ball.position(at: far.end, along: y) => "0 m", ball.position(at: high.start, along: y) => "0 m"13  deduce_from "at most", ball.direction(at: far.start) => "45 deg"14  deduce_from "How high", ball.direction(at: high.start) => "90 deg"15  deduce_from "How high", ball.velocity(at: high.end, along: y) => "0 m/s"16  deduce_from "the same ball", ball.speed(at: high.start) => ball.speed(at: far.start)17  height = ball.position(at: high.end, along: y)18  answer height:, in: "m"19  conclusion "The farthest throw, at forty-five degrees, and the highest, straight up, both " \20             "depend on the square of the speed over the pull, and the height is always half the " \21             "range."22end

playground.rb

1require_relative "../solutions/as_high_as_it_goes_far"23play_with :as_high_as_it_goes_far do4  title "As High as It Goes Far"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 ball during far along x	$Δ x_{x} (f)$	m	—
position of ball at end of far along x	$x_{x} (f₁)$	m	—
position of ball at start of far along x	$x_{x} (f₀)$	m	—
displacement of ball during far along y	$Δ x_{y} (f)$	m	—
position of ball at end of far along y	$x_{y} (f₁)$	m	—
position of ball at start of far along y	$x_{y} (f₀)$	m	—
displacement of ball during high along x	$Δ x_{x} (h)$	m	—
position of ball at end of high along x	$x_{x} (h₁)$	m	—
position of ball at start of high along x	$x_{x} (h₀)$	m	—
displacement of ball during high along y	$Δ x_{y} (h)$	m	—
position of ball at end of high along y	$x_{y} (h₁)$	m	—
position of ball at start of high along y	$x_{y} (h₀)$	m	—
average velocity of ball during far along x	$v_{av,x} (f)$	m/s	—
time taken of far	$Δ t_{f}$	s	0 … 1e+05
average velocity of ball during far along y	$v_{av,y} (f)$	m/s	—
average velocity of ball during high along x	$v_{av,x} (h)$	m/s	—
time taken of high	$Δ t_{h}$	s	0 … 1e+05
average velocity of ball during high along y	$v_{av,y} (h)$	m/s	—
average speed of ball during far	$c_{av} (f)$	m/s	—
path length of ball during far	$d (f)$	m	—
average speed of ball during high	$c_{av} (h)$	m/s	—
path length of ball during high	$d (h)$	m	—
change of velocity of ball during far along x	$Δ v_{x} (f)$	m/s	—
velocity of ball at end of far from ground along x	$v_{x} (f₁)$	m/s	—
velocity of ball at start of far from ground along x	$v_{x} (f₀)$	m/s	—
change of velocity of ball during far along y	$Δ v_{y} (f)$	m/s	—
velocity of ball at end of far from ground along y	$v_{y} (f₁)$	m/s	—
velocity of ball at start of far from ground along y	$v_{y} (f₀)$	m/s	—
change of velocity of ball during high along x	$Δ v_{x} (h)$	m/s	—
velocity of ball at end of high from ground along x	$v_{x} (h₁)$	m/s	—
velocity of ball at start of high from ground along x	$v_{x} (h₀)$	m/s	—
change of velocity of ball during high along y	$Δ v_{y} (h)$	m/s	—
velocity of ball at end of high from ground along y	$v_{y} (h₁)$	m/s	—
velocity of ball at start of high from ground along y	$v_{y} (h₀)$	m/s	—
average acceleration of ball during far along x	$a_{av,x} (f)$	m/s²	—
average acceleration of ball during far along y	$a_{av,y} (f)$	m/s²	—
average acceleration of ball during high along x	$a_{av,x} (h)$	m/s²	—
average acceleration of ball during high along y	$a_{av,y} (h)$	m/s²	—
acceleration of ball during far along x	$a_{x} (f)$	m/s²	—
acceleration of ball during far along y	$a_{y} (f)$	m/s²	—
acceleration of ball during high along x	$a_{x} (h)$	m/s²	—
acceleration of ball during high along y	$a_{y} (h)$	m/s²	—
velocity of ground at start of far from ground along x	$v_{g,x} (f₀)$	m/s	—
velocity of ground at start of far from ground	$v_{g} (f₀)$	m/s	—
direction of ground at start of far from ground	$θ_{g} (f₀)$	—	-3.142 … 3.142
velocity of ground at start of far from ground along y	$v_{g,y} (f₀)$	m/s	—
velocity of ground at end of far from ground along x	$v_{g,x} (f₁)$	m/s	—
velocity of ground at end of far from ground	$v_{g} (f₁)$	m/s	—
direction of ground at end of far from ground	$θ_{g} (f₁)$	—	-3.142 … 3.142
velocity of ground at end of far from ground along y	$v_{g,y} (f₁)$	m/s	—
velocity of ground at start of high from ground along x	$v_{g,x} (h₀)$	m/s	—
velocity of ground at start of high from ground	$v_{g} (h₀)$	m/s	—
direction of ground at start of high from ground	$θ_{g} (h₀)$	—	-3.142 … 3.142
velocity of ground at start of high from ground along y	$v_{g,y} (h₀)$	m/s	—
velocity of ground at end of high from ground along x	$v_{g,x} (h₁)$	m/s	—
velocity of ground at end of high from ground	$v_{g} (h₁)$	m/s	—
direction of ground at end of high from ground	$θ_{g} (h₁)$	—	-3.142 … 3.142
velocity of ground at end of high from ground along y	$v_{g,y} (h₁)$	m/s	—
velocity of ball at start of far from ground	$v (f₀)$	m/s	—
direction of ball at start of far from ground	$θ (f₀)$	—	-3.142 … 3.142
velocity of ball at end of far from ground	$v (f₁)$	m/s	—
direction of ball at end of far from ground	$θ (f₁)$	—	-3.142 … 3.142
velocity of ball at start of high from ground	$v (h₀)$	m/s	—
direction of ball at start of high from ground	$θ (h₀)$	—	-3.142 … 3.142
velocity of ball at end of high from ground	$v (h₁)$	m/s	—
direction of ball at end of high from ground	$θ (h₁)$	—	-3.142 … 3.142
displacement of ball during far	$Δ x (f)$	m	—
position of ball at end of far	$x (f₁)$	m	—
position of ball at start of far	$x (f₀)$	m	—
displacement of ball during high	$Δ x (h)$	m	—
position of ball at end of high	$x (h₁)$	m	—
position of ball at start of high	$x (h₀)$	m	—
average velocity of ball during far	$v_{av} (f)$	m/s	—
average velocity of ball during high	$v_{av} (h)$	m/s	—
change of velocity of ball during far	$Δ v (f)$	m/s	—
change of velocity of ball during high	$Δ v (h)$	m/s	—
average acceleration of ball during far	$a_{av} (f)$	m/s²	—
average acceleration of ball during high	$a_{av} (h)$	m/s²	—
acceleration of ball during far	$a (f)$	m/s²	—
acceleration of ball during high	$a (h)$	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} (f₁) - x_{x} (f₀)

change of position(far, x)

Δ x_{y} (f) = x_{y} (f₁) - x_{y} (f₀)

change of position(far, y)

Δ x_{x} (h) = x_{x} (h₁) - x_{x} (h₀)

change of position(high, x)

Δ x_{y} (h) = x_{y} (h₁) - x_{y} (h₀)

change of position(high, y)

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

displacement a time(far, x)

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

displacement a time(far, y)

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

displacement a time(high, x)

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

displacement a time(high, y)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(far, x)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(far, y)

c_{av} (h) = \frac{d (h)}{Δ t_{h}}

path a time(high, x)

c_{av} (h) = \frac{d (h)}{Δ t_{h}}

path a time(high, y)

Δ v_{x} (f) = v_{x} (f₁) - v_{x} (f₀)

velocity gained(far, x)

Δ v_{y} (f) = v_{y} (f₁) - v_{y} (f₀)

velocity gained(far, y)

Δ v_{x} (h) = v_{x} (h₁) - v_{x} (h₀)

velocity gained(high, x)

Δ v_{y} (h) = v_{y} (h₁) - v_{y} (h₀)

velocity gained(high, y)

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

velocity gained a time(far, x)

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

velocity gained a time(far, y)

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

velocity gained a time(high, x)

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

velocity gained a time(high, y)

v_{x} (f₁) = v_{x} (f₀) + a_{x} (f) \cdot Δ t_{f}

velocity after a time(far, x)

v_{y} (f₁) = v_{y} (f₀) + a_{y} (f) \cdot Δ t_{f}

velocity after a time(far, y)

v_{x} (h₁) = v_{x} (h₀) + a_{x} (h) \cdot Δ t_{h}

velocity after a time(high, x)

v_{y} (h₁) = v_{y} (h₀) + a_{y} (h) \cdot Δ t_{h}

velocity after a time(high, y)

x_{x} (f₁) = x_{x} (f₀) + v_{x} (f₀) \cdot Δ t_{f} + \frac{a_{x} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(far, x)

x_{y} (f₁) = x_{y} (f₀) + v_{y} (f₀) \cdot Δ t_{f} + \frac{a_{y} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(far, y)

x_{x} (h₁) = x_{x} (h₀) + v_{x} (h₀) \cdot Δ t_{h} + \frac{a_{x} (h) \cdot {Δ t_{h}}^{2}}{2}

position after a time(high, x)

x_{y} (h₁) = x_{y} (h₀) + v_{y} (h₀) \cdot Δ t_{h} + \frac{a_{y} (h) \cdot {Δ t_{h}}^{2}}{2}

position after a time(high, y)

{v_{x} (f₁)}^{2} = {v_{x} (f₀)}^{2} + 2 \cdot a_{x} (f) \cdot (x_{x} (f₁) - x_{x} (f₀))

velocity after a distance(far, x)

{v_{y} (f₁)}^{2} = {v_{y} (f₀)}^{2} + 2 \cdot a_{y} (f) \cdot (x_{y} (f₁) - x_{y} (f₀))

velocity after a distance(far, y)

{v_{x} (h₁)}^{2} = {v_{x} (h₀)}^{2} + 2 \cdot a_{x} (h) \cdot (x_{x} (h₁) - x_{x} (h₀))

velocity after a distance(high, x)

{v_{y} (h₁)}^{2} = {v_{y} (h₀)}^{2} + 2 \cdot a_{y} (h) \cdot (x_{y} (h₁) - x_{y} (h₀))

velocity after a distance(high, y)

x_{x} (f₁) - x_{x} (f₀) = \frac{(v_{x} (f₁) + v_{x} (f₀)) \cdot Δ t_{f}}{2}

mean of the velocities(far, x)

x_{y} (f₁) - x_{y} (f₀) = \frac{(v_{y} (f₁) + v_{y} (f₀)) \cdot Δ t_{f}}{2}

mean of the velocities(far, y)

x_{x} (h₁) - x_{x} (h₀) = \frac{(v_{x} (h₁) + v_{x} (h₀)) \cdot Δ t_{h}}{2}

mean of the velocities(high, x)

x_{y} (h₁) - x_{y} (h₀) = \frac{(v_{y} (h₁) + v_{y} (h₀)) \cdot Δ t_{h}}{2}

mean of the velocities(high, y)

v_{g,x} (f₀) = v_{g} (f₀) \cdot \cos θ_{g} (f₀)

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

v_{g,y} (f₀) = v_{g} (f₀) \cdot \sin θ_{g} (f₀)

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

v_{g,x} (f₁) = v_{g} (f₁) \cdot \cos θ_{g} (f₁)

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

v_{g,y} (f₁) = v_{g} (f₁) \cdot \sin θ_{g} (f₁)

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

v_{g,x} (h₀) = v_{g} (h₀) \cdot \cos θ_{g} (h₀)

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

v_{g,y} (h₀) = v_{g} (h₀) \cdot \sin θ_{g} (h₀)

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

v_{g,x} (h₁) = v_{g} (h₁) \cdot \cos θ_{g} (h₁)

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

v_{g,y} (h₁) = v_{g} (h₁) \cdot \sin θ_{g} (h₁)

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

v_{x} (f₀) = v (f₀) \cdot \cos θ (f₀)

parts of a velocity(ball, start of far, x)

v_{y} (f₀) = v (f₀) \cdot \sin θ (f₀)

parts of a velocity(ball, start of far, y)

v_{x} (f₁) = v (f₁) \cdot \cos θ (f₁)

parts of a velocity(ball, end of far, x)

v_{y} (f₁) = v (f₁) \cdot \sin θ (f₁)

parts of a velocity(ball, end of far, y)

v_{x} (h₀) = v (h₀) \cdot \cos θ (h₀)

parts of a velocity(ball, start of high, x)

v_{y} (h₀) = v (h₀) \cdot \sin θ (h₀)

parts of a velocity(ball, start of high, y)

v_{x} (h₁) = v (h₁) \cdot \cos θ (h₁)

parts of a velocity(ball, end of high, x)

v_{y} (h₁) = v (h₁) \cdot \sin θ (h₁)

parts of a velocity(ball, end of high, y)

a_{x} (f) = 0

falling freely(far, x)

a_{y} (f) = - g

falling freely(far, y)

a_{x} (h) = 0

falling freely(high, x)

a_{y} (h) = - g

falling freely(high, y)

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

magnitude from components(displacement of ball during far)

x (f₁) = \sqrt{{x_{x} (f₁)}^{2} + {x_{y} (f₁)}^{2}}

magnitude from components(position of ball at end of far)

x (f₀) = \sqrt{{x_{x} (f₀)}^{2} + {x_{y} (f₀)}^{2}}

magnitude from components(position of ball at start of far)

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

magnitude from components(displacement of ball during high)

x (h₁) = \sqrt{{x_{x} (h₁)}^{2} + {x_{y} (h₁)}^{2}}

magnitude from components(position of ball at end of high)

x (h₀) = \sqrt{{x_{x} (h₀)}^{2} + {x_{y} (h₀)}^{2}}

magnitude from components(position of ball at start of high)

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

magnitude from components(average velocity of ball during far)

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

magnitude from components(average velocity of ball during high)

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

magnitude from components(change of velocity of ball during far)

v (f₁) = \sqrt{{v_{x} (f₁)}^{2} + {v_{y} (f₁)}^{2}}

magnitude from components(velocity of ball at end of far from ground)

v (f₀) = \sqrt{{v_{x} (f₀)}^{2} + {v_{y} (f₀)}^{2}}

magnitude from components(velocity of ball at start of far from ground)

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

magnitude from components(change of velocity of ball during high)

v (h₁) = \sqrt{{v_{x} (h₁)}^{2} + {v_{y} (h₁)}^{2}}

magnitude from components(velocity of ball at end of high from ground)

v (h₀) = \sqrt{{v_{x} (h₀)}^{2} + {v_{y} (h₀)}^{2}}

magnitude from components(velocity of ball at start of high from ground)

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

magnitude from components(average acceleration of ball during far)

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

magnitude from components(average acceleration of ball during high)

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

magnitude from components(acceleration of ball during far)

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

magnitude from components(acceleration of ball during high)

v_{g} (f₀) = \sqrt{{v_{g,x} (f₀)}^{2} + {v_{g,y} (f₀)}^{2}}

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

v_{g} (f₁) = \sqrt{{v_{g,x} (f₁)}^{2} + {v_{g,y} (f₁)}^{2}}

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

v_{g} (h₀) = \sqrt{{v_{g,x} (h₀)}^{2} + {v_{g,y} (h₀)}^{2}}

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

v_{g} (h₁) = \sqrt{{v_{g,x} (h₁)}^{2} + {v_{g,y} (h₁)}^{2}}

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

v (h₀) = v (f₀)

the same ball