A Raindrop Held Back

Q. A drop of mass 1.00 g falls from a height of 1.00 km under gravity, taken as 10 m/s², and an opposing resistive force. It hits the ground with speed 50.0 m/s. What is the work done by the unknown resistive force, in joules? expects -8.75 J

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The web of concepts

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The equations it used

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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?

The playground

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}

The solution

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 Ruby

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

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

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

The laws

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)