Riding on a Speeding Train

solved NewtonsSecondLaw

Q. A stone of mass 0.1 kg $m_{s} = 0.1$ kg is lying on the floor of a train which is accelerating with 1 m/s² $a_{t} (s) = 1$ m/s², the stone being at rest relative to the train $a_{s} (c) = a_{t} (s)$ . What is the magnitude of the net force acting on the stone, in newtons?

Inputs

mass

m_{s}

0.1 kg

train acceleration

a_{t}

1 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
$a_{s} (c) = a_{t} (s)$ from “at rest relative to the train”

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

Working

at rest relative to the train $a_{s} (c) = a_{t} (s)$
$a_{s} (c) = 1$
$a_{s} (c) = 1$ m/s²
second law(carried, x) $F_{net,s} (c) = m_{s} \cdot a_{s} (c)$
$F_{net,s} (c) = 0.1 \cdot 1$
$F_{net,s} (c) = 0.1$ N
magnitude from components(net force on stone during carried) $| F_{net,s} (c) | = \sqrt{{F_{net,s} (c)}^{2}}$
$| F_{net,s} (c) | = \sqrt{{0.1}^{2}}$
$| F_{net,s} (c) | = 0.1$ N

Conclusion

The stone keeps up with the train, so something must speed it up as the train speeds up: the floor pulls it along by friction, and that is the whole of the net force.

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 :riding_on_a_speeding_train do4  kind :numerical5  describes "A stone of mass %{mass} is lying on the floor of a train which is accelerating with " \6            "%{train_acceleration}, the stone being at rest relative to the train. %{force}, in newtons?"7  given mass: "0.1 kg"8  given train_acceleration: "1 m/s2"9  asks force: "What is the magnitude of the net force acting on the stone"10end

solution.rb

1require_relative "../questions/riding_on_a_speeding_train"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :riding_on_a_speeding_train, using: NewtonsSecondLaw do5  objects :stone, :train6  axes :x7  train.goes(:speeding_up)8  stone.goes(:carried)910  take mass: stone.mass11  take train_acceleration: train.acceleration1213  deduce_from "at rest relative to the train", stone.acceleration => train.acceleration1415  force = stone.net_force.magnitude16  answer force:, in: "N"1718  conclusion "The stone keeps up with the train, so something must speed it up as the train speeds up: the " \19             "floor pulls it along by friction, and that is the whole of the net force."20end

playground.rb

1require_relative "../solutions/riding_on_a_speeding_train"23play_with :riding_on_a_speeding_train do4  title "Riding on 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 x	$Δ x_{t} (s)$	m	—
position of train at end of speeding up along x	$x_{t} (s₁)$	m	—
position of train at start of speeding up along x	$x_{t} (s₀)$	m	—
displacement of stone during carried along x	$Δ x_{s} (c)$	m	—
position of stone at end of carried along x	$x_{s} (c₁)$	m	—
position of stone at start of carried along x	$x_{s} (c₀)$	m	—
average velocity of train during speeding up along x	$v_{av,t} (s)$	m/s	—
time taken of speeding up	$Δ t_{s}$	s	0 … 1e+05
average velocity of stone during carried along x	$v_{av,s} (c)$	m/s	—
time taken of carried	$Δ t_{c}$	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 carried	$c_{av,s} (c)$	m/s	—
path length of stone during carried	$d_{s} (c)$	m	—
change of velocity of train during speeding up along x	$Δ v_{t} (s)$	m/s	—
velocity of train at end of speeding up from ground along x	$v_{t} (s₁)$	m/s	—
velocity of train at start of speeding up from ground along x	$v_{t} (s₀)$	m/s	—
change of velocity of stone during carried along x	$Δ v_{s} (c)$	m/s	—
velocity of stone at end of carried from ground along x	$v_{s} (c₁)$	m/s	—
velocity of stone at start of carried from ground along x	$v_{s} (c₀)$	m/s	—
average acceleration of train during speeding up along x	$a_{av,t} (s)$	m/s²	—
average acceleration of stone during carried along x	$a_{av,s} (c)$	m/s²	—
acceleration of train during speeding up along x	$a_{t} (s)$	m/s²	—
acceleration of stone during carried along x	$a_{s} (c)$	m/s²	—
net force on train during speeding up along x	$F_{net,t} (s)$	N	—
mass of train	$m_{t}$	kg	0 … ∞
net force on stone during carried along x	$F_{net,s} (c)$	N	—
mass of stone	$m_{s}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on stone	$\| W_{s} \|$	N	—
weight on train	$\| W_{t} \|$	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 carried	$\| Δ x_{s} (c) \|$	m	—
position of stone at end of carried	$\| x_{s} (c₁) \|$	m	—
position of stone at start of carried	$\| x_{s} (c₀) \|$	m	—
average velocity of train during speeding up	$\| v_{av,t} (s) \|$	m/s	—
average velocity of stone during carried	$\| v_{av,s} (c) \|$	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 carried	$\| Δ v_{s} (c) \|$	m/s	—
velocity of stone at end of carried from ground	$\| v_{s} (c₁) \|$	m/s	—
velocity of stone at start of carried from ground	$\| v_{s} (c₀) \|$	m/s	—
average acceleration of train during speeding up	$\| a_{av,t} (s) \|$	m/s²	—
average acceleration of stone during carried	$\| a_{av,s} (c) \|$	m/s²	—
acceleration of train during speeding up	$\| a_{t} (s) \|$	m/s²	—
acceleration of stone during carried	$\| a_{s} (c) \|$	m/s²	—
net force on train during speeding up	$\| F_{net,t} (s) \|$	N	—
net force on stone during carried	$\| F_{net,s} (c) \|$	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, x)

Δ x_{s} (c) = x_{s} (c₁) - x_{s} (c₀)

change of position(carried, x)

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

displacement a time(speeding up, x)

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

displacement a time(carried, x)

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

path a time(speeding up, x)

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

path a time(carried, x)

Δ v_{t} (s) = v_{t} (s₁) - v_{t} (s₀)

velocity gained(speeding up, x)

Δ v_{s} (c) = v_{s} (c₁) - v_{s} (c₀)

velocity gained(carried, x)

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

velocity gained a time(speeding up, x)

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

velocity gained a time(carried, x)

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

velocity after a time(speeding up, x)

v_{s} (c₁) = v_{s} (c₀) + a_{s} (c) \cdot Δ t_{c}

velocity after a time(carried, x)

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, x)

x_{s} (c₁) = x_{s} (c₀) + v_{s} (c₀) \cdot Δ t_{c} + \frac{a_{s} (c) \cdot {Δ t_{c}}^{2}}{2}

position after a time(carried, x)

{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, x)

{v_{s} (c₁)}^{2} = {v_{s} (c₀)}^{2} + 2 \cdot a_{s} (c) \cdot (x_{s} (c₁) - x_{s} (c₀))

velocity after a distance(carried, x)

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

mean of the velocities(speeding up, x)

x_{s} (c₁) - x_{s} (c₀) = \frac{(v_{s} (c₁) + v_{s} (c₀)) \cdot Δ t_{c}}{2}

mean of the velocities(carried, x)

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

second law(speeding up, x)

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

second law(carried, x)

| 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} (c) | = \sqrt{{Δ x_{s} (c)}^{2}}

magnitude from components(displacement of stone during carried)

| x_{s} (c₁) | = \sqrt{{x_{s} (c₁)}^{2}}

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

| x_{s} (c₀) | = \sqrt{{x_{s} (c₀)}^{2}}

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

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

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

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

magnitude from components(average velocity of stone during carried)

| 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} (c) | = \sqrt{{Δ v_{s} (c)}^{2}}

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

| v_{s} (c₁) | = \sqrt{{v_{s} (c₁)}^{2}}

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

| v_{s} (c₀) | = \sqrt{{v_{s} (c₀)}^{2}}

magnitude from components(velocity of stone at start of carried 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} (c) | = \sqrt{{a_{av,s} (c)}^{2}}

magnitude from components(average acceleration of stone during carried)

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

magnitude from components(acceleration of train during speeding up)

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

magnitude from components(acceleration of stone during carried)

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

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

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

magnitude from components(net force on stone during carried)

a_{s} (c) = a_{t} (s)

at rest relative to the train

m_{s} \neq 0

has mass?(stone)

m_{t} \neq 0

has mass?(train)

F_{net,s} (c) = 0

free of forces?(stone, carried)

F_{net,t} (s) = 0

free of forces?(train, speeding up)