As High as It Goes Far

Q. A cricketer can throw a ball at most 100 m. How high can they throw the same ball, in metres? expects 50.00 m

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

Inputs

farthest

x_{x}

100.00 m

Outputs

height

x_{y}

The solution

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 Ruby

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

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

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

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