A Splash Heard from a Tower

solved UniformAcceleration

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?

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}

How it's solved

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 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_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

solution.rb

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

playground.rb

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 it drew on

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