Dropped From a Speeding Train

solved NewtonsSecondLaw

Q. A stone of mass 0.1 kg $m_{s} = 0.1$ kg is dropped from the window of a train accelerating with 1 m/s² $| a_{t} (s) | = 1$ m/s². Neglect air resistance $F_{sa} (f) = 0$ N, and take g as $W_{s} = - | W_{s} |$ 10 m/s² $g = 10$ m/s². What is the magnitude of the net force acting on the stone just after it is dropped, in newtons?

Inputs

mass

m_{s}

0.1 kg

train acceleration

| a_{t} |

1 m/s²

gravity

g

10 m/s²

Outputs

force

| F_{net,s} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m_{s} = 0.1$ kg
$| a_{t} (s) | = 1$ m/s²
$g = 10$ m/s²
$F_{sa} (f) = 0$ N from “Neglect air resistance”
$W_{s} = - | W_{s} |$ from “take g as”

Find $| F_{net,s} (f) |$ .

Working

weight of a body(stone) $| W_{s} | = m_{s} \cdot g$
$| W_{s} | = 0.1 \cdot 10$
$| W_{s} | = 1$ N
take g as $W_{s} = - | W_{s} |$
$W_{s} = - 1$
$W_{s} = −1$ N
net force is the resultant(stone, falling, y) $F_{net,s} (f) = F_{sa} (f) + W_{s}$
$F_{net,s} (f) = 0 + (−1)$
$F_{net,s} (f) = −1$ N
magnitude from components(net force on stone during falling) $| F_{net,s} (f) | = \sqrt{{F_{net,s} (f)}^{2}}$
$| F_{net,s} (f) | = \sqrt{{(−1)}^{2}}$
$| F_{net,s} (f) | = 1$ N

Conclusion

Once the stone leaves the window the train no longer pushes it, so the only force left is its weight; how the train is moving makes no difference.

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 :dropped_from_a_speeding_train do4  kind :numerical5  describes "A stone of mass %{mass} is dropped from the window of a train accelerating with %{train_acceleration}. " \6            "Neglect air resistance, and take g as %{gravity}. %{force} just after it is dropped, in newtons?"7  given mass: "0.1 kg"8  given train_acceleration: "1 m/s2"9  given gravity: "10 m/s2"10  asks force: "What is the magnitude of the net force acting on the stone"11end

solution.rb

1require_relative "../questions/dropped_from_a_speeding_train"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :dropped_from_a_speeding_train, using: NewtonsSecondLaw do5  objects :stone, :train, :air6  axes :y7  train.goes(:speeding_up)8  stone.goes(:falling)9  deduce_from "take g as", stone.weight.direction => -y1011  take mass: stone.mass12  take train_acceleration: train.acceleration.magnitude13  take gravity: g1415  deduce_from "Neglect air resistance", stone.force(by: air) => "0 N"1617  force = stone.net_force.magnitude18  answer force:, in: "N"1920  conclusion "Once the stone leaves the window the train no longer pushes it, so the only force left is its " \21             "weight; how the train is moving makes no difference."22end

playground.rb

1require_relative "../solutions/dropped_from_a_speeding_train"23play_with :dropped_from_a_speeding_train do4  title "Dropped From a Speeding Train"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	$work$	—	—	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	$F$	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
displacement of train during speeding up along y	$Δ x_{t} (s)$	m	—
position of train at end of speeding up along y	$x_{t} (s₁)$	m	—
position of train at start of speeding up along y	$x_{t} (s₀)$	m	—
displacement of stone during falling along y	$Δ x_{s} (f)$	m	—
position of stone at end of falling along y	$x_{s} (f₁)$	m	—
position of stone at start of falling along y	$x_{s} (f₀)$	m	—
average velocity of train during speeding up along y	$v_{av,t} (s)$	m/s	—
time taken of speeding up	$Δ t_{s}$	s	0 … 1e+05
average velocity of stone during falling along y	$v_{av,s} (f)$	m/s	—
time taken of falling	$Δ t_{f}$	s	0 … 1e+05
average speed of train during speeding up	$c_{av,t} (s)$	m/s	—
path length of train during speeding up	$d_{t} (s)$	m	—
average speed of stone during falling	$c_{av,s} (f)$	m/s	—
path length of stone during falling	$d_{s} (f)$	m	—
change of velocity of train during speeding up along y	$Δ v_{t} (s)$	m/s	—
velocity of train at end of speeding up from ground along y	$v_{t} (s₁)$	m/s	—
velocity of train at start of speeding up from ground along y	$v_{t} (s₀)$	m/s	—
change of velocity of stone during falling along y	$Δ v_{s} (f)$	m/s	—
velocity of stone at end of falling from ground along y	$v_{s} (f₁)$	m/s	—
velocity of stone at start of falling from ground along y	$v_{s} (f₀)$	m/s	—
average acceleration of train during speeding up along y	$a_{av,t} (s)$	m/s²	—
average acceleration of stone during falling along y	$a_{av,s} (f)$	m/s²	—
acceleration of train during speeding up along y	$a_{t} (s)$	m/s²	—
acceleration of stone during falling along y	$a_{s} (f)$	m/s²	—
net force on train during speeding up along y	$F_{net,t} (s)$	N	—
mass of train	$m_{t}$	kg	0 … ∞
net force on stone during falling along y	$F_{net,s} (f)$	N	—
mass of stone	$m_{s}$	kg	0 … ∞
mass of air	$m_{a}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on stone	$\| W_{s} \|$	N	—
weight on train	$\| W_{t} \|$	N	—
weight on air	$\| W_{a} \|$	N	—
force on stone by air during falling	$\| F_{sa} (f) \|$	N	—
weight on stone along y	$W_{s}$	N	—
force on stone by air during falling along y	$F_{sa} (f)$	N	—
displacement of train during speeding up	$\| Δ x_{t} (s) \|$	m	—
position of train at end of speeding up	$\| x_{t} (s₁) \|$	m	—
position of train at start of speeding up	$\| x_{t} (s₀) \|$	m	—
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	—
average velocity of train during speeding up	$\| v_{av,t} (s) \|$	m/s	—
average velocity of stone during falling	$\| v_{av,s} (f) \|$	m/s	—
change of velocity of train during speeding up	$\| Δ v_{t} (s) \|$	m/s	—
velocity of train at end of speeding up from ground	$\| v_{t} (s₁) \|$	m/s	—
velocity of train at start of speeding up from ground	$\| v_{t} (s₀) \|$	m/s	—
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	—
average acceleration of train during speeding up	$\| a_{av,t} (s) \|$	m/s²	—
average acceleration of stone during falling	$\| a_{av,s} (f) \|$	m/s²	—
acceleration of train during speeding up	$\| a_{t} (s) \|$	m/s²	—
acceleration of stone during falling	$\| a_{s} (f) \|$	m/s²	—
net force on train during speeding up	$\| F_{net,t} (s) \|$	N	—
net force on stone during falling	$\| F_{net,s} (f) \|$	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_{t} (s) = x_{t} (s₁) - x_{t} (s₀)

change of position(speeding up, y)

Δ x_{s} (f) = x_{s} (f₁) - x_{s} (f₀)

change of position(falling, y)

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

displacement a time(speeding up, y)

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

displacement a time(falling, y)

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

path a time(speeding up, y)

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

path a time(falling, y)

Δ v_{t} (s) = v_{t} (s₁) - v_{t} (s₀)

velocity gained(speeding up, y)

Δ v_{s} (f) = v_{s} (f₁) - v_{s} (f₀)

velocity gained(falling, y)

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

velocity gained a time(speeding up, y)

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

velocity gained a time(falling, y)

v_{t} (s₁) = v_{t} (s₀) + a_{t} (s) \cdot Δ t_{s}

velocity after a time(speeding up, y)

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

velocity after a time(falling, y)

x_{t} (s₁) = x_{t} (s₀) + v_{t} (s₀) \cdot Δ t_{s} + \frac{a_{t} (s) \cdot {Δ t_{s}}^{2}}{2}

position after a time(speeding up, y)

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, y)

{v_{t} (s₁)}^{2} = {v_{t} (s₀)}^{2} + 2 \cdot a_{t} (s) \cdot (x_{t} (s₁) - x_{t} (s₀))

velocity after a distance(speeding up, y)

{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, y)

x_{t} (s₁) - x_{t} (s₀) = \frac{(v_{t} (s₁) + v_{t} (s₀)) \cdot Δ t_{s}}{2}

mean of the velocities(speeding up, y)

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

mean of the velocities(falling, y)

F_{net,t} (s) = m_{t} \cdot a_{t} (s)

second law(speeding up, y)

F_{net,s} (f) = m_{s} \cdot a_{s} (f)

second law(falling, y)

| W_{s} | = m_{s} \cdot g

weight of a body(stone)

F_{net,s} (f) = F_{sa} (f) + W_{s}

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

| Δ x_{t} (s) | = \sqrt{{Δ x_{t} (s)}^{2}}

magnitude from components(displacement of train during speeding up)

| x_{t} (s₁) | = \sqrt{{x_{t} (s₁)}^{2}}

magnitude from components(position of train at end of speeding up)

| x_{t} (s₀) | = \sqrt{{x_{t} (s₀)}^{2}}

magnitude from components(position of train at start of speeding up)

| Δ x_{s} (f) | = \sqrt{{Δ x_{s} (f)}^{2}}

magnitude from components(displacement of stone during falling)

| x_{s} (f₁) | = \sqrt{{x_{s} (f₁)}^{2}}

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

| x_{s} (f₀) | = \sqrt{{x_{s} (f₀)}^{2}}

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

| v_{av,t} (s) | = \sqrt{{v_{av,t} (s)}^{2}}

magnitude from components(average velocity of train during speeding up)

| v_{av,s} (f) | = \sqrt{{v_{av,s} (f)}^{2}}

magnitude from components(average velocity of stone during falling)

| Δ v_{t} (s) | = \sqrt{{Δ v_{t} (s)}^{2}}

magnitude from components(change of velocity of train during speeding up)

| v_{t} (s₁) | = \sqrt{{v_{t} (s₁)}^{2}}

magnitude from components(velocity of train at end of speeding up from ground)

| v_{t} (s₀) | = \sqrt{{v_{t} (s₀)}^{2}}

magnitude from components(velocity of train at start of speeding up from ground)

| Δ v_{s} (f) | = \sqrt{{Δ v_{s} (f)}^{2}}

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

| v_{s} (f₁) | = \sqrt{{v_{s} (f₁)}^{2}}

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

| v_{s} (f₀) | = \sqrt{{v_{s} (f₀)}^{2}}

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

| a_{av,t} (s) | = \sqrt{{a_{av,t} (s)}^{2}}

magnitude from components(average acceleration of train during speeding up)

| a_{av,s} (f) | = \sqrt{{a_{av,s} (f)}^{2}}

magnitude from components(average acceleration of stone during falling)

| a_{t} (s) | = \sqrt{{a_{t} (s)}^{2}}

magnitude from components(acceleration of train during speeding up)

| a_{s} (f) | = \sqrt{{a_{s} (f)}^{2}}

magnitude from components(acceleration of stone during falling)

| F_{net,t} (s) | = \sqrt{{F_{net,t} (s)}^{2}}

magnitude from components(net force on train during speeding up)

| F_{net,s} (f) | = \sqrt{{F_{net,s} (f)}^{2}}

magnitude from components(net force on stone during falling)

| W_{s} | = \sqrt{{W_{s}}^{2}}

magnitude from components(weight on stone)

| F_{sa} (f) | = \sqrt{{F_{sa} (f)}^{2}}

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

W_{s} = - | W_{s} |

take g as

m_{s} \neq 0

has mass?(stone)

m_{t} \neq 0

has mass?(train)

m_{a} \neq 0

has mass?(air)

F_{net,s} (f) = 0

free of forces?(stone, falling)

F_{net,t} (s) = 0

free of forces?(train, speeding up)