A Raindrop Held Back

solved KineticEnergy

Q. A drop of mass 1.00 g $m = 1.00$ g falls $v_{1} = - | v_{1} |$ from a height of 1.00 km $x_{0} = 1.00$ km under gravity $W = - | W |$ , taken as 10 m/s² $g = 10$ m/s², and an opposing resistive force $F_{a} = | F_{a} |$ . It hits the ground $x_{1} = 0$ m with speed 50.0 m/s $| v_{1} | = 50.0$ m/s. What is the work done by the unknown resistive force, in joules?

Inputs

mass

m

1 g

height

x_{0}

1 km

gravity

g

10 m/s²

speed

| v_{1} |

50.0 m/s

Outputs

work

W_{F,a}

How it's solved

Solved with KineticEnergy

Given

$m = 1.00$ g
$x_{0} = 1.00$ km
$g = 10$ m/s²
$| v_{1} | = 50.0$ m/s
$x_{1} = 0$ m from “hits the ground”
$W = - | W |$ from “under gravity”
$v_{1} = - | v_{1} |$ from “falls”
$F_{a} = | F_{a} |$ from “an opposing resistive force”

Assumed

$v_{0} = 0$ m/s

Find $W_{F,a}$ .

Working

$m = 1.00$ g $= 0.001$ kg
$x_{0} = 1.00$ km $= 1000$ m
falls $v_{1} = - | v_{1} |$
$v_{1} = - 50$
$v_{1} = −50$ m/s
velocity after a distance(falling, y) ${v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})$
${(−50)}^{2} = 0^{2} + 2 \cdot a \cdot (0 - 1000)$
$a = −1.25$ m/s²
second law(falling, y) $F_{net} = m \cdot a$
$F_{net} = 0.001 \cdot (−1.25)$
$F_{net} = −0.00125$ N
weight of a body(drop) $| W | = m \cdot g$
$| W | = 0.001 \cdot 10$
$| W | = 0.01$ N
under gravity $W = - | W |$
$W = - 0.01$
$W = −0.01$ N
net force is the resultant(drop, falling, y) $F_{net} = F_{a} + W$
$(−0.00125) = F_{a} + (−0.01)$
$F_{a} = 0.00875$ N
change of position(falling, y) $Δ x = x_{1} - x_{0}$
$Δ x = 0 - 1000$
$Δ x = −1000$ m
work of a steady force(drop, falling, y) $W_{F,a} = F_{a} \cdot Δ x$
$W_{F,a} = 0.00875 \cdot (−1000)$
$W_{F,a} = −8.75$ J
$W_{F,a} \approx −8.75$ Jto 3 significant figures

Conclusion

Gravity puts in mgh, far more than the drop keeps as kinetic energy at the ground; the air takes out the difference, so its work is negative.

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_raindrop_held_back do4  kind :numerical5  describes "A drop of mass %{mass} falls from a height of %{height} under gravity, taken as %{gravity}, and an " \6            "opposing resistive force. It hits the ground with speed %{speed}. %{work}, in joules?"7  given mass: "1.00 g"8  given height: "1.00 km"9  given gravity: "10 m/s2"10  given speed: "50.0 m/s"11  asks work: "What is the work done by the unknown resistive force"12end

solution.rb

1require_relative "../questions/a_raindrop_held_back"2require_relative "../science/work_energy_and_power/kinetic_energy"34solve :a_raindrop_held_back, using: KineticEnergy do5  objects :drop, :air6  axes :y7  drop.goes(:falling)8  deduce_from "under gravity", drop.weight.direction => -y910  take mass: drop.mass11  take height: drop.initial_position12  take gravity: g13  take speed: drop.final_velocity.magnitude1415  assume drop.initial_velocity => "0 m/s"16  deduce_from "hits the ground", drop.final_position => "0 m"17  deduce_from "falls", drop.final_velocity.direction => -y18  deduce_from "an opposing resistive force", drop.force(by: air).direction => y1920  work = drop.work(by: air)21  answer work:, in: "J", figures: 32223  conclusion "Gravity puts in mgh, far more than the drop keeps as kinetic energy at the ground; the " \24             "air takes out the difference, so its work is negative."25end

playground.rb

1require_relative "../solutions/a_raindrop_held_back"23play_with :a_raindrop_held_back do4  title "A Raindrop Held Back"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$W_{F}$	J	—	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	$F$	N	—	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
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
work by gravity	$W_{g}$	J	—	Work
net work	$W_{net}$	J	—	Work
kinetic energy	$K$	J	—	KineticEnergy
change of kinetic energy	$Δ K$	J	—	KineticEnergy
displacement of drop during falling along y	$Δ x$	m	—
position of drop at end of falling along y	$x_{1}$	m	—
position of drop at start of falling along y	$x_{0}$	m	—
average velocity of drop during falling along y	$v_{av}$	m/s	—
time taken of falling	$Δ t$	s	0 … 1e+05
average speed of drop during falling	$c_{av}$	m/s	—
path length of drop during falling	$d$	m	—
change of velocity of drop during falling along y	$Δ v$	m/s	—
velocity of drop at end of falling from ground along y	$v_{1}$	m/s	—
velocity of drop at start of falling from ground along y	$v_{0}$	m/s	—
average acceleration of drop during falling along y	$a_{av}$	m/s²	—
acceleration of drop during falling along y	$a$	m/s²	—
net force on drop during falling along y	$F_{net}$	N	—
mass of drop	$m$	kg	0 … ∞
mass of air	$m_{a}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on drop	$\| W \|$	N	—
weight on air	$\| W_{a} \|$	N	—
force on drop by air during falling	$\| F_{a} \|$	N	—
weight on drop along y	$W$	N	—
force on drop by air during falling along y	$F_{a}$	N	—
work on drop by air during falling	$W_{F,a}$	J	—
net work on drop during falling	$W_{net}$	J	—
work by gravity on drop during falling	$W_{g}$	J	—
kinetic energy of drop at start of falling	$K_{0}$	J	—
kinetic energy of drop at end of falling	$K_{1}$	J	—
change of kinetic energy of drop during falling	$Δ K$	J	—
displacement of drop during falling	$\| Δ x \|$	m	—
position of drop at end of falling	$\| x_{1} \|$	m	—
position of drop at start of falling	$\| x_{0} \|$	m	—
average velocity of drop during falling	$\| v_{av} \|$	m/s	—
change of velocity of drop during falling	$\| Δ v \|$	m/s	—
velocity of drop at end of falling from ground	$\| v_{1} \|$	m/s	—
velocity of drop at start of falling from ground	$\| v_{0} \|$	m/s	—
average acceleration of drop during falling	$\| a_{av} \|$	m/s²	—
acceleration of drop during falling	$\| a \|$	m/s²	—
net force on drop during falling	$\| F_{net} \|$	N	—

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

m_{b} \neq 0

has mass?

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_{1} - x_{0}

change of position(falling, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(falling, y)

c_{av} = \frac{d}{Δ t}

path a time(falling, y)

Δ v = v_{1} - v_{0}

velocity gained(falling, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(falling, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(falling, y)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(falling, y)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(falling, y)

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

mean of the velocities(falling, y)

F_{net} = m \cdot a

second law(falling, y)

| W | = m \cdot g

weight of a body(drop)

F_{net} = F_{a} + W

net force is the resultant(drop, falling, y)

W_{F,a} = F_{a} \cdot Δ x

work of a steady force(drop, falling, y)

W_{net} = F_{net} \cdot Δ x

work of the net force(drop, falling)

W_{g} = W \cdot Δ x

work of the weight(drop, falling)

K_{0} = \frac{m \cdot {v_{0}}^{2}}{2}

kinetic energy of a body(drop, falling, 1)

K_{1} = \frac{m \cdot {v_{1}}^{2}}{2}

kinetic energy of a body(drop, falling, 2)

Δ K = K_{1} - K_{0}

kinetic energy gained(drop, falling)

K_{1} - K_{0} = W_{F,a} + W_{g}

work energy theorem(drop, falling)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of drop during falling)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of drop at end of falling)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of drop at start of falling)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of drop during falling)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of drop during falling)

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of drop at end of falling from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of drop at start of falling from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of drop during falling)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of drop during falling)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on drop during falling)

| W | = \sqrt{W^{2}}

magnitude from components(weight on drop)

| F_{a} | = \sqrt{{F_{a}}^{2}}

magnitude from components(force on drop by air during falling)

W = - | W |

under gravity

v_{1} = - | v_{1} |

falls

F_{a} = | F_{a} |

an opposing resistive force

m \neq 0

has mass?(drop)

m_{a} \neq 0

has mass?(air)

F_{net} = 0

free of forces?(drop, falling)