Under the Ceiling

solved Projectile

Q. A ball is thrown at 40 m/s $v (thrown) = 40$ m/s in a long hall whose ceiling is 25 m $x_{y} (peak) = 25$ m high. Taking g as 9.8 m/s² $g = 9.8$ m/s², how far can it go without touching the ceiling $v_{y} (peak) = 0$ m/s, in metres?

Inputs

speed

v

40.0 m/s

gravity

g

9.8 m/s²

ceiling

x_{y}

25.00 m

Outputs

distance

x_{x}

How it's solved

Solved with Projectile

Given

$v (thrown) = 40$ m/s
$g = 9.8$ m/s²
$x_{y} (peak) = 25$ m
$v_{y} (peak) = 0$ m/s from “without touching the ceiling”
$v_{y} (thrown)$ : the positive one of the values that fit from “without touching the ceiling”
$θ (thrown)$ : the smallest of the values that fit from “how far can it go”

Assumed

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

Find $x_{x} (lands)$ .

Working

falling freely(up, y) $a_{y} (u) = - g$
$a_{y} (u) = - 9.8$
$a_{y} (u) = −9.8$ m/s²
velocity after a distance(up, y) ${v_{y} (peak)}^{2} = {v_{y} (thrown)}^{2} + 2 \cdot a_{y} (u) \cdot (x_{y} (peak) - x_{y} (thrown))$
$0^{2} = {v_{y} (thrown)}^{2} + 2 \cdot (−9.8) \cdot (25 - 0)$
$v_{y} (thrown) = 22.1359436$ m/s
parts of a velocity(ball, thrown, y) $v_{y} (thrown) = v (thrown) \cdot \sin θ (thrown)$
$22.1359436 = 40 \cdot \sin θ (thrown)$
$θ (thrown) = 33.6 °, 146.4 °$ the smallest is taken
$θ (thrown) = 33.60°$
parts of a velocity(ball, thrown, x) $v_{x} (thrown) = v (thrown) \cdot \cos θ (thrown)$
$v_{x} (thrown) = 40 \cdot \cos 33.6 °$
$v_{x} (thrown) = 33.3166625$ m/s
velocity after a time(up, y) $v_{y} (peak) = v_{y} (thrown) + a_{y} (u) \cdot Δ t_{u}$
$0 = 22.1359436 + (−9.8) \cdot Δ t_{u}$
$Δ t_{u} = 2.25876976$ s
falling freely(up, x) $a_{x} (u) = 0$
$a_{x} (u) = 0$ m/s²
position after a time(up, x) $x_{x} (peak) = x_{x} (thrown) + v_{x} (thrown) \cdot Δ t_{u} + \frac{a_{x} (u) \cdot {Δ t_{u}}^{2}}{2}$
$x_{x} (peak) = 0 + 33.3166625 \cdot 2.25876976 + \frac{0 \cdot {2.25876976}^{2}}{2}$
$x_{x} (peak) = 75.2546697$ m
velocity after a time(up, x) $v_{x} (peak) = v_{x} (thrown) + a_{x} (u) \cdot Δ t_{u}$
$v_{x} (peak) = 33.3166625 + 0 \cdot 2.25876976$
$v_{x} (peak) = 33.3166625$ m/s
falling freely(down, y) $a_{y} (d) = - g$
$a_{y} (d) = - 9.8$
$a_{y} (d) = −9.8$ m/s²
position after a time(down, y) $x_{y} (lands) = x_{y} (peak) + v_{y} (peak) \cdot Δ t_{d} + \frac{a_{y} (d) \cdot {Δ t_{d}}^{2}}{2}$
$0 = 25 + 0 \cdot Δ t_{d} + \frac{(−9.8) \cdot {Δ t_{d}}^{2}}{2}$
$Δ t_{d} = 2.25876976$ s
falling freely(down, x) $a_{x} (d) = 0$
$a_{x} (d) = 0$ m/s²
position after a time(down, x) $x_{x} (lands) = x_{x} (peak) + v_{x} (peak) \cdot Δ t_{d} + \frac{a_{x} (d) \cdot {Δ t_{d}}^{2}}{2}$
$x_{x} (lands) = 75.2546697 + 33.3166625 \cdot 2.25876976 + \frac{0 \cdot {2.25876976}^{2}}{2}$
$x_{x} (lands) = 150.51 m$

Conclusion

The ceiling caps how much of the speed can go upward; throw at the angle that just grazes it, and the rest of the speed carries the ball as far as it can.

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 :under_the_ceiling do4  kind :numerical5  describes "A ball is thrown at %{speed} in a long hall whose ceiling is %{ceiling} high. Taking g as %{gravity}, " \6          "%{distance}, in metres?"7  given speed: "40 m/s"8  given gravity: "9.8 m/s2"9  given ceiling: "25 m"10  asks distance: "how far can it go without touching the ceiling"11end

solution.rb

1require_relative "../questions/under_the_ceiling"2require_relative "../science/motion_in_a_plane/projectile"34solve :under_the_ceiling, using: Projectile do5  objects :ball6  axes :x, :y7  instants :thrown, :peak, :lands8  ball.goes(:up, during: thrown..peak)9  ball.goes(:down, during: peak..lands)10  take speed: ball.speed(at: thrown), gravity: g, ceiling: ball.position(at: peak, along: y)11  assume ball.position(at: thrown, along: x) => "0 m", ball.position(at: thrown, along: y) => "0 m",12         ball.position(at: lands, along: y) => "0 m"13  deduce_from "without touching the ceiling", ball.velocity(at: peak, along: y) => "0 m/s",14                                              ball.velocity(at: thrown, along: y) => :positive15  deduce_from "how far can it go", ball.direction(at: thrown) => :smallest16  distance = ball.position(at: lands, along: x)17  answer distance:, in: "m"18  conclusion "The ceiling caps how much of the speed can go upward; throw at the angle that just " \19             "grazes it, and the rest of the speed carries the ball as far as it can."20end

playground.rb

1require_relative "../solutions/under_the_ceiling"23play_with :under_the_ceiling do4  title "Under the Ceiling"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	$Δ x_{x} (u)$	m	—
position of ball at peak along x	$x_{x} (peak)$	m	—
position of ball at thrown along x	$x_{x} (thrown)$	m	—
displacement of ball during up along y	$Δ x_{y} (u)$	m	—
position of ball at peak along y	$x_{y} (peak)$	m	—
position of ball at thrown along y	$x_{y} (thrown)$	m	—
displacement of ball during down along x	$Δ x_{x} (d)$	m	—
position of ball at lands along x	$x_{x} (lands)$	m	—
displacement of ball during down along y	$Δ x_{y} (d)$	m	—
position of ball at lands along y	$x_{y} (lands)$	m	—
average velocity of ball during up along x	$v_{av,x} (u)$	m/s	—
time taken of up	$Δ t_{u}$	s	0 … 1e+05
average velocity of ball during up along y	$v_{av,y} (u)$	m/s	—
average velocity of ball during down along x	$v_{av,x} (d)$	m/s	—
time taken of down	$Δ t_{d}$	s	0 … 1e+05
average velocity of ball during down along y	$v_{av,y} (d)$	m/s	—
average speed of ball during up	$c_{av} (u)$	m/s	—
path length of ball during up	$d (u)$	m	—
average speed of ball during down	$c_{av} (d)$	m/s	—
path length of ball during down	$d (d)$	m	—
change of velocity of ball during up along x	$Δ v_{x} (u)$	m/s	—
velocity of ball at peak from ground along x	$v_{x} (peak)$	m/s	—
velocity of ball at thrown from ground along x	$v_{x} (thrown)$	m/s	—
change of velocity of ball during up along y	$Δ v_{y} (u)$	m/s	—
velocity of ball at peak from ground along y	$v_{y} (peak)$	m/s	—
velocity of ball at thrown from ground along y	$v_{y} (thrown)$	m/s	—
change of velocity of ball during down along x	$Δ v_{x} (d)$	m/s	—
velocity of ball at lands from ground along x	$v_{x} (lands)$	m/s	—
change of velocity of ball during down along y	$Δ v_{y} (d)$	m/s	—
velocity of ball at lands from ground along y	$v_{y} (lands)$	m/s	—
average acceleration of ball during up along x	$a_{av,x} (u)$	m/s²	—
average acceleration of ball during up along y	$a_{av,y} (u)$	m/s²	—
average acceleration of ball during down along x	$a_{av,x} (d)$	m/s²	—
average acceleration of ball during down along y	$a_{av,y} (d)$	m/s²	—
acceleration of ball during up along x	$a_{x} (u)$	m/s²	—
acceleration of ball during up along y	$a_{y} (u)$	m/s²	—
acceleration of ball during down along x	$a_{x} (d)$	m/s²	—
acceleration of ball during down along y	$a_{y} (d)$	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 peak from ground along x	$v_{g,x} (peak)$	m/s	—
velocity of ground at peak from ground	$v_{g} (peak)$	m/s	—
direction of ground at peak from ground	$θ_{g} (peak)$	—	-3.142 … 3.142
velocity of ground at peak from ground along y	$v_{g,y} (peak)$	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 ball at thrown from ground	$v (thrown)$	m/s	—
direction of ball at thrown from ground	$θ (thrown)$	—	-3.142 … 3.142
velocity of ball at peak from ground	$v (peak)$	m/s	—
direction of ball at peak from ground	$θ (peak)$	—	-3.142 … 3.142
velocity of ball at lands from ground	$v (lands)$	m/s	—
direction of ball at lands from ground	$θ (lands)$	—	-3.142 … 3.142
displacement of ball during up	$Δ x (u)$	m	—
position of ball at peak	$x (peak)$	m	—
position of ball at thrown	$x (thrown)$	m	—
displacement of ball during down	$Δ x (d)$	m	—
position of ball at lands	$x (lands)$	m	—
average velocity of ball during up	$v_{av} (u)$	m/s	—
average velocity of ball during down	$v_{av} (d)$	m/s	—
change of velocity of ball during up	$Δ v (u)$	m/s	—
change of velocity of ball during down	$Δ v (d)$	m/s	—
average acceleration of ball during up	$a_{av} (u)$	m/s²	—
average acceleration of ball during down	$a_{av} (d)$	m/s²	—
acceleration of ball during up	$a (u)$	m/s²	—
acceleration of ball during down	$a (d)$	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} (peak) - x_{x} (thrown)

change of position(up, x)

Δ x_{y} (u) = x_{y} (peak) - x_{y} (thrown)

change of position(up, y)

Δ x_{x} (d) = x_{x} (lands) - x_{x} (peak)

change of position(down, x)

Δ x_{y} (d) = x_{y} (lands) - x_{y} (peak)

change of position(down, 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)

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)

Δ v_{x} (u) = v_{x} (peak) - v_{x} (thrown)

velocity gained(up, x)

Δ v_{y} (u) = v_{y} (peak) - v_{y} (thrown)

velocity gained(up, y)

Δ v_{x} (d) = v_{x} (lands) - v_{x} (peak)

velocity gained(down, x)

Δ v_{y} (d) = v_{y} (lands) - v_{y} (peak)

velocity gained(down, 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)

v_{x} (peak) = v_{x} (thrown) + a_{x} (u) \cdot Δ t_{u}

velocity after a time(up, x)

v_{y} (peak) = v_{y} (thrown) + a_{y} (u) \cdot Δ t_{u}

velocity after a time(up, y)

v_{x} (lands) = v_{x} (peak) + a_{x} (d) \cdot Δ t_{d}

velocity after a time(down, x)

v_{y} (lands) = v_{y} (peak) + a_{y} (d) \cdot Δ t_{d}

velocity after a time(down, y)

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

position after a time(up, x)

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

position after a time(up, y)

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

position after a time(down, x)

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

position after a time(down, y)

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

velocity after a distance(up, x)

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

velocity after a distance(up, y)

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

velocity after a distance(down, x)

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

velocity after a distance(down, y)

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

mean of the velocities(up, x)

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

mean of the velocities(up, y)

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

mean of the velocities(down, x)

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

mean of the velocities(down, 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} (peak) = v_{g} (peak) \cdot \cos θ_{g} (peak)

parts of a velocity(ground, peak, x)

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

parts of a velocity(ground, peak, 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_{x} (thrown) = v (thrown) \cdot \cos θ (thrown)

parts of a velocity(ball, thrown, x)

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

parts of a velocity(ball, thrown, y)

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

parts of a velocity(ball, peak, x)

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

parts of a velocity(ball, peak, y)

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

parts of a velocity(ball, lands, x)

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

parts of a velocity(ball, lands, 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)

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

magnitude from components(displacement of ball during up)

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

magnitude from components(position of ball at peak)

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

magnitude from components(position of ball at thrown)

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

magnitude from components(displacement of ball during down)

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

magnitude from components(position of ball at lands)

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

magnitude from components(average velocity of ball during up)

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

magnitude from components(average velocity of ball during down)

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

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

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

magnitude from components(velocity of ball at peak from ground)

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

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

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

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

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

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

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

magnitude from components(average acceleration of ball during up)

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

magnitude from components(average acceleration of ball during down)

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

magnitude from components(acceleration of ball during up)

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

magnitude from components(acceleration of ball during down)

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

magnitude from components(velocity of ground at peak 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)