Dropped From a Speeding Train

Q. A stone of mass 0.1 kg is dropped from the window of a train accelerating with 1 m/s². Neglect air resistance, and take g as 10 m/s². What is the magnitude of the net force acting on the stone just after it is dropped, in newtons? expects 1 N

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 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?

The playground

Inputs

mass

m_{s}

0.1 kg

train acceleration

| a_{t} |

1 m/s²

gravity

g

10 m/s²

Outputs

force

| F_{net,s} |

The solution

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 Ruby

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

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

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

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)