A Splash Heard from a Tower

Q. A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m/s (g = 9.8 m/s²), in seconds? expects 8.7 s

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 stone dropped $v_{s} (f₀) = 0$ m/s, $a_{s} (f) = - g$ from the top of a tower of height 300 m $x_{S} (r₁) = 300$ m splashes into the water $x_{s} (f₁) = 0$ m, $x_{S} (r₀) = 0$ m of a pond near the base of the tower. When is the splash heard at the top $Δ t_{w} = Δ t_{f} + Δ t_{r}$ given that the speed of sound in air $a_{S} (r) = 0$ m/s² is 340 m/s $v_{S} (r₀) = 340$ m/s (g = 9.8 m/s² $g = 9.8$ m/s²), in seconds?

The playground

Inputs

height

x_{S}

300.00 m

speed

v_{S}

340.0 m/s

gravity

g

9.8 m/s²

Outputs

time

Δ t_{w}

The solution

Solved with UniformAcceleration

Given

$x_{S} (r₁) = 300$ m
$v_{S} (r₀) = 340$ m/s
$g = 9.8$ m/s²
$v_{s} (f₀) = 0$ m/s from “A stone dropped”
$x_{s} (f₁) = 0$ m from “splashes into the water”
$x_{S} (r₀) = 0$ m from “splashes into the water”
$a_{S} (r) = 0$ m/s² from “the speed of sound in air”
$a_{s} (f) = - g$ from “A stone dropped”
$x_{s} (f₀) = x_{S} (r₁)$ from “dropped from the top”
$Δ t_{w} = Δ t_{f} + Δ t_{r}$ from “heard at the top”

Find $Δ t_{w}$ .

Working

dropped from the top $x_{s} (f₀) = x_{S} (r₁)$
$x_{s} (f₀) = 300$
$x_{s} (f₀) = 300$ m
A stone dropped $a_{s} (f) = - g$
$a_{s} (f) = - 9.8$
$a_{s} (f) = −9.8$ m/s²
position after a time(falling) $x_{s} (f₁) = x_{s} (f₀) + v_{s} (f₀) \cdot Δ t_{f} + \frac{a_{s} (f) \cdot {Δ t_{f}}^{2}}{2}$
$0 = 300 + 0 \cdot Δ t_{f} + \frac{(−9.8) \cdot {Δ t_{f}}^{2}}{2}$
$Δ t_{f} = 7.82460796$ s
position after a time(rising) $x_{S} (r₁) = x_{S} (r₀) + v_{S} (r₀) \cdot Δ t_{r} + \frac{a_{S} (r) \cdot {Δ t_{r}}^{2}}{2}$
$300 = 0 + 340 \cdot Δ t_{r} + \frac{0 \cdot {Δ t_{r}}^{2}}{2}$
$Δ t_{r} = 0.882352941$ s
heard at the top $Δ t_{w} = Δ t_{f} + Δ t_{r}$
$Δ t_{w} = 7.82460796 + 0.882352941$
$Δ t_{w} = 8.71 s$
$Δ t_{w} \approx 8.7$ sto 2 significant figures

Conclusion

The stone takes most of the time, falling from rest under gravity; the sound then climbs back at its steady speed, adding a little under a second.

The Ruby

1require_relative "../physics"23question :a_splash_heard_from_a_tower do4  kind :numerical5  describes "A stone dropped from the top of a tower of height %{height} splashes into the water of a pond near the " \6            "base of the tower. %{time} given that the speed of sound in air is %{speed} (g = %{gravity}), in " \7            "seconds?"8  given height: "300 m"9  given speed: "340 m/s"10  given gravity: "9.8 m/s2"11  asks time: "When is the splash heard at the top"12end

1require_relative "../questions/a_splash_heard_from_a_tower"2require_relative "../science/motion_in_a_straight_line/uniform_acceleration"34solve :a_splash_heard_from_a_tower, using: UniformAcceleration do5  objects :stone, :listener, sound: :S6  stone.goes(:falling)7  sound.goes(:rising)8  listener.goes(:waiting)910  take height: sound.final_position11  take speed: sound.initial_velocity12  take gravity: g1314  deduce_from "A stone dropped", stone.initial_velocity => "0 m/s", stone.acceleration => -g15  deduce_from "splashes into the water", stone.final_position => "0 m", sound.initial_position => "0 m"16  deduce_from "dropped from the top", stone.initial_position => sound.final_position17  deduce_from "heard at the top", waiting.time_taken => falling.time_taken + rising.time_taken18  deduce_from "the speed of sound in air", sound.acceleration => "0 m/s2"1920  time = waiting.time_taken21  answer time:, in: "s", figures: 22223  conclusion "The stone takes most of the time, falling from rest under gravity; the sound then climbs back at " \24             "its steady speed, adding a little under a second."25end

1require_relative "../solutions/a_splash_heard_from_a_tower"23play_with :a_splash_heard_from_a_tower do4  title "A Splash Heard from a Tower"5end

The laws

quantity	written	units	within	from
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
velocity	$v$	m/s	—	Acceleration
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
acceleration	$a$	m/s²	—	UniformAcceleration
gravity	$g$	m/s²	—	UniformAcceleration
time	$t$	s	0 … 1e+05	UniformAcceleration
displacement of stone during falling	$Δ x_{s} (f)$	m	—
position of stone at end of falling	$x_{s} (f₁)$	m	—
position of stone at start of falling	$x_{s} (f₀)$	m	—
displacement of sound during rising	$Δ x_{S} (r)$	m	—
position of sound at end of rising	$x_{S} (r₁)$	m	—
position of sound at start of rising	$x_{S} (r₀)$	m	—
displacement of listener during waiting	$Δ x_{l} (w)$	m	—
position of listener at end of waiting	$x_{l} (w₁)$	m	—
position of listener at start of waiting	$x_{l} (w₀)$	m	—
average velocity of stone during falling	$v_{av,s} (f)$	m/s	—
time taken of falling	$Δ t_{f}$	s	0 … 1e+05
average velocity of sound during rising	$v_{av,S} (r)$	m/s	—
time taken of rising	$Δ t_{r}$	s	0 … 1e+05
average velocity of listener during waiting	$v_{av,l} (w)$	m/s	—
time taken of waiting	$Δ t_{w}$	s	0 … 1e+05
average speed of stone during falling	$c_{av,s} (f)$	m/s	—
path length of stone during falling	$d_{s} (f)$	m	—
average speed of sound during rising	$c_{av,S} (r)$	m/s	—
path length of sound during rising	$d_{S} (r)$	m	—
average speed of listener during waiting	$c_{av,l} (w)$	m/s	—
path length of listener during waiting	$d_{l} (w)$	m	—
change of velocity of stone during falling	$Δ v_{s} (f)$	m/s	—
velocity of stone at end of falling from ground	$v_{s} (f₁)$	m/s	—
velocity of stone at start of falling from ground	$v_{s} (f₀)$	m/s	—
change of velocity of sound during rising	$Δ v_{S} (r)$	m/s	—
velocity of sound at end of rising from ground	$v_{S} (r₁)$	m/s	—
velocity of sound at start of rising from ground	$v_{S} (r₀)$	m/s	—
change of velocity of listener during waiting	$Δ v_{l} (w)$	m/s	—
velocity of listener at end of waiting from ground	$v_{l} (w₁)$	m/s	—
velocity of listener at start of waiting from ground	$v_{l} (w₀)$	m/s	—
average acceleration of stone during falling	$a_{av,s} (f)$	m/s²	—
average acceleration of sound during rising	$a_{av,S} (r)$	m/s²	—
average acceleration of listener during waiting	$a_{av,l} (w)$	m/s²	—
acceleration of stone during falling	$a_{s} (f)$	m/s²	—
acceleration of sound during rising	$a_{S} (r)$	m/s²	—
acceleration of listener during waiting	$a_{l} (w)$	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

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

Δ x_{s} (f) = x_{s} (f₁) - x_{s} (f₀)

change of position(falling)

Δ x_{S} (r) = x_{S} (r₁) - x_{S} (r₀)

change of position(rising)

Δ x_{l} (w) = x_{l} (w₁) - x_{l} (w₀)

change of position(waiting)

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

displacement a time(falling)

v_{av,S} (r) = \frac{Δ x_{S} (r)}{Δ t_{r}}

displacement a time(rising)

v_{av,l} (w) = \frac{Δ x_{l} (w)}{Δ t_{w}}

displacement a time(waiting)

c_{av,s} (f) = \frac{d_{s} (f)}{Δ t_{f}}

path a time(falling)

c_{av,S} (r) = \frac{d_{S} (r)}{Δ t_{r}}

path a time(rising)

c_{av,l} (w) = \frac{d_{l} (w)}{Δ t_{w}}

path a time(waiting)

Δ v_{s} (f) = v_{s} (f₁) - v_{s} (f₀)

velocity gained(falling)

Δ v_{S} (r) = v_{S} (r₁) - v_{S} (r₀)

velocity gained(rising)

Δ v_{l} (w) = v_{l} (w₁) - v_{l} (w₀)

velocity gained(waiting)

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

velocity gained a time(falling)

a_{av,S} (r) = \frac{Δ v_{S} (r)}{Δ t_{r}}

velocity gained a time(rising)

a_{av,l} (w) = \frac{Δ v_{l} (w)}{Δ t_{w}}

velocity gained a time(waiting)

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

velocity after a time(falling)

v_{S} (r₁) = v_{S} (r₀) + a_{S} (r) \cdot Δ t_{r}

velocity after a time(rising)

v_{l} (w₁) = v_{l} (w₀) + a_{l} (w) \cdot Δ t_{w}

velocity after a time(waiting)

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

position after a time(falling)

x_{S} (r₁) = x_{S} (r₀) + v_{S} (r₀) \cdot Δ t_{r} + \frac{a_{S} (r) \cdot {Δ t_{r}}^{2}}{2}

position after a time(rising)

x_{l} (w₁) = x_{l} (w₀) + v_{l} (w₀) \cdot Δ t_{w} + \frac{a_{l} (w) \cdot {Δ t_{w}}^{2}}{2}

position after a time(waiting)

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

velocity after a distance(falling)

{v_{S} (r₁)}^{2} = {v_{S} (r₀)}^{2} + 2 \cdot a_{S} (r) \cdot (x_{S} (r₁) - x_{S} (r₀))

velocity after a distance(rising)

{v_{l} (w₁)}^{2} = {v_{l} (w₀)}^{2} + 2 \cdot a_{l} (w) \cdot (x_{l} (w₁) - x_{l} (w₀))

velocity after a distance(waiting)

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

mean of the velocities(falling)

x_{S} (r₁) - x_{S} (r₀) = \frac{(v_{S} (r₁) + v_{S} (r₀)) \cdot Δ t_{r}}{2}

mean of the velocities(rising)

x_{l} (w₁) - x_{l} (w₀) = \frac{(v_{l} (w₁) + v_{l} (w₀)) \cdot Δ t_{w}}{2}

mean of the velocities(waiting)

a_{s} (f) = - g

A stone dropped

x_{s} (f₀) = x_{S} (r₁)

dropped from the top

Δ t_{w} = Δ t_{f} + Δ t_{r}

heard at the top