Riding on a Speeding Train

Q. A stone of mass 0.1 kg is lying on the floor of a train which is accelerating with 1 m/s², the stone being at rest relative to the train. What is the magnitude of the net force acting on the stone, in newtons? expects 0.1 N

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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?

The playground

Inputs

mass

m_{s}

0.1 kg

train acceleration

a_{t}

1 m/s²

Outputs

force

| F_{net,s} |

The solution

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 Ruby

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

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

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

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)