A Cricket Ball Rises

solved Projectile

Q. A cricket ball is thrown at 28 m/s $v_{0} = 28$ m/s, 30° $θ_{0} = 30 °$ above the horizontal. Taking g as 9.8 m/s² $g = 9.8$ m/s², how high does it rise $v_{1,y} = 0$ m/s, in metres?

Inputs

speed

v_{0}

28.0 m/s

angle

θ_{0}

30.00°

gravity

g

9.8 m/s²

Outputs

height

x_{1,y}

How it's solved

Solved with Projectile

Given

$v_{0} = 28$ m/s
$θ_{0} = 30 °$
$g = 9.8$ m/s²
$v_{1,y} = 0$ m/s from “how high does it rise”

Assumed

$x_{0,y} = 0$ m

Find $x_{1,y}$ .

Working

parts of a velocity(ball, start of up, y) $v_{0,y} = v_{0} \cdot \sin θ_{0}$
$v_{0,y} = 28 \cdot \sin 30 °$
$v_{0,y} = 14$ m/s
falling freely(up, y) $a_{y} = - g$
$a_{y} = - 9.8$
$a_{y} = −9.8$ m/s²
velocity after a time(up, y) $v_{1,y} = v_{0,y} + a_{y} \cdot Δ t$
$0 = 14 + (−9.8) \cdot Δ t$
$Δ t = 1.42857143$ s
position after a time(up, y) $x_{1,y} = x_{0,y} + v_{0,y} \cdot Δ t + \frac{a_{y} \cdot {Δ t}^{2}}{2}$
$x_{1,y} = 0 + 14 \cdot 1.42857143 + \frac{(−9.8) \cdot {1.42857143}^{2}}{2}$
$x_{1,y} = 10.00 m$

Conclusion

Only the upward part of the throw fights gravity, so the height is set by that part alone.

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 :a_cricket_ball_rises do4  kind :numerical5  describes "A cricket ball is thrown at %{speed}, %{angle} above the horizontal. Taking g as %{gravity}, %{height}, in metres?"6  given speed: "28 m/s"7  given angle: "30 deg"8  given gravity: "9.8 m/s2"9  asks height: "how high does it rise"10end

solution.rb

1require_relative "../questions/a_cricket_ball_rises"2require_relative "../science/motion_in_a_plane/projectile"34solve :a_cricket_ball_rises, using: Projectile do5  objects :ball6  axes :x, :y7  ball.goes(:up)8  take speed: ball.initial_speed, angle: ball.initial_direction, gravity: g9  assume ball.initial_position(along: y) => "0 m"10  deduce_from "how high does it rise", ball.final_velocity(along: y) => "0 m/s"11  height = ball.final_position(along: y)12  answer height:, in: "m"13  conclusion "Only the upward part of the throw fights gravity, so the height is set by that part " \14             "alone."15end

playground.rb

1require_relative "../solutions/a_cricket_ball_rises"23play_with :a_cricket_ball_rises do4  title "A Cricket Ball Rises"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 up along x	$Δ x_{x}$	m	—
position of ball at end of up along x	$x_{1,x}$	m	—
position of ball at start of up along x	$x_{0,x}$	m	—
displacement of ball during up along y	$Δ x_{y}$	m	—
position of ball at end of up along y	$x_{1,y}$	m	—
position of ball at start of up along y	$x_{0,y}$	m	—
average velocity of ball during up along x	$v_{av,x}$	m/s	—
time taken of up	$Δ t$	s	0 … 1e+05
average velocity of ball during up along y	$v_{av,y}$	m/s	—
average speed of ball during up	$c_{av}$	m/s	—
path length of ball during up	$d$	m	—
change of velocity of ball during up along x	$Δ v_{x}$	m/s	—
velocity of ball at end of up from ground along x	$v_{1,x}$	m/s	—
velocity of ball at start of up from ground along x	$v_{0,x}$	m/s	—
change of velocity of ball during up along y	$Δ v_{y}$	m/s	—
velocity of ball at end of up from ground along y	$v_{1,y}$	m/s	—
velocity of ball at start of up from ground along y	$v_{0,y}$	m/s	—
average acceleration of ball during up along x	$a_{av,x}$	m/s²	—
average acceleration of ball during up along y	$a_{av,y}$	m/s²	—
acceleration of ball during up along x	$a_{x}$	m/s²	—
acceleration of ball during up along y	$a_{y}$	m/s²	—
velocity of ground at start of up from ground along x	$v_{0,g,x}$	m/s	—
velocity of ground at start of up from ground	$v_{0,g}$	m/s	—
direction of ground at start of up from ground	$θ_{0,g}$	—	-3.142 … 3.142
velocity of ground at start of up from ground along y	$v_{0,g,y}$	m/s	—
velocity of ground at end of up from ground along x	$v_{1,g,x}$	m/s	—
velocity of ground at end of up from ground	$v_{1,g}$	m/s	—
direction of ground at end of up from ground	$θ_{1,g}$	—	-3.142 … 3.142
velocity of ground at end of up from ground along y	$v_{1,g,y}$	m/s	—
velocity of ball at start of up from ground	$v_{0}$	m/s	—
direction of ball at start of up from ground	$θ_{0}$	—	-3.142 … 3.142
velocity of ball at end of up from ground	$v_{1}$	m/s	—
direction of ball at end of up from ground	$θ_{1}$	—	-3.142 … 3.142
displacement of ball during up	$Δ x$	m	—
position of ball at end of up	$x_{1}$	m	—
position of ball at start of up	$x_{0}$	m	—
average velocity of ball during up	$v_{av}$	m/s	—
change of velocity of ball during up	$Δ v$	m/s	—
average acceleration of ball during up	$a_{av}$	m/s²	—
acceleration of ball during up	$a$	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(up, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(up, y)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(up, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(up, y)

c_{av} = \frac{d}{Δ t}

path a time(up, x)

c_{av} = \frac{d}{Δ t}

path a time(up, y)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(up, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(up, y)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(up, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(up, y)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(up, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(up, 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(up, 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(up, y)

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

velocity after a distance(up, x)

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

velocity after a distance(up, y)

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

mean of the velocities(up, x)

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

mean of the velocities(up, y)

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

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

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

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

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

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

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

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

v_{0,x} = v_{0} \cdot \cos θ_{0}

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

v_{0,y} = v_{0} \cdot \sin θ_{0}

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

v_{1,x} = v_{1} \cdot \cos θ_{1}

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

v_{1,y} = v_{1} \cdot \sin θ_{1}

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

a_{x} = 0

falling freely(up, x)

a_{y} = - g

falling freely(up, y)

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

magnitude from components(displacement of ball during up)

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

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

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

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

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

magnitude from components(average velocity of ball during up)

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

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

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

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

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

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

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

magnitude from components(average acceleration of ball during up)

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

magnitude from components(acceleration of ball during up)

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

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