Under the Ceiling

Q. A ball is thrown at 40 m/s in a long hall whose ceiling is 25 m high. Taking g as 9.8 m/s², how far can it go without touching the ceiling, in metres? expects 150.51 m

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The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

Inputs

speed

v

40.0 m/s

gravity

g

9.8 m/s²

ceiling

x_{y}

25.00 m

Outputs

distance

x_{x}

The solution

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 Ruby

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

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

1require_relative "../solutions/under_the_ceiling"23play_with :under_the_ceiling do4  title "Under the Ceiling"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 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)