Before the Push Back

solved NewtonsSecondLaw

Q. A body of mass 0.40 kg $m = 0.40$ kg moving initially with a constant speed $F_{net} (c) = 0$ N of 10 m/s $v (applied) = 10$ m/s to the north is subject to a constant force of 8.0 N $| F_{p} (p) | = 8.0$ N directed towards the south $F_{p} (p) = - | F_{p} (p) |$ for 30 s. Take the instant the force is applied to be t = 0, and the position of the body at that time to be x = 0 $x (applied) = 0$ m. What was its position 5 s $Δ t_{c} = 5$ s before the force is applied, at t = –5 s, in metres?

Inputs

mass

m

0.4 kg

speed

v

10.0 m/s

force

| F_{p} |

8 N

earlier

Δ t_{c}

5.00 s

Outputs

position

x

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 0.40$ kg
$v (applied) = 10$ m/s
$| F_{p} (p) | = 8.0$ N
$Δ t_{c} = 5$ s
$x (applied) = 0$ m from “x = 0”
$F_{net} (c) = 0$ N from “constant speed”
$F_{p} (p) = - | F_{p} (p) |$ from “directed towards the south”

Find $x (earlier_on)$ .

Working

second law(coasting, x) $F_{net} (c) = m \cdot a (c)$
$0 = 0.4 \cdot a (c)$
$a (c) = 0$ m/s²
velocity after a time(coasting, x) $v (applied) = v (earlier_on) + a (c) \cdot Δ t_{c}$
$10 = v (earlier_on) + 0 \cdot 5$
$v (earlier_on) = 10$ m/s
position after a time(coasting, x) $x (applied) = x (earlier_on) + v (earlier_on) \cdot Δ t_{c} + \frac{a (c) \cdot {Δ t_{c}}^{2}}{2}$
$0 = x (earlier_on) + 10 \cdot 5 + \frac{0 \cdot 5^{2}}{2}$
$x (earlier_on) = -50.00 m$

Conclusion

Before the force acts the body just runs on at its steady speed, so the force has no say in where it was.

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 :before_the_push_back do4  kind :numerical5  describes "A body of mass %{mass} moving initially with a constant speed of %{speed} to the north is subject to a " \6            "constant force of %{force} directed towards the south for 30 s. Take the instant the force is " \7            "applied to be t = 0, and the position of the body at that time to be x = 0. %{position} %{earlier} " \8            "before the force is applied, at t = –5 s, in metres?"9  given mass: "0.40 kg"10  given speed: "10 m/s"11  given force: "8.0 N"12  given earlier: "5 s"13  asks position: "What was its position"14end

solution.rb

1require_relative "../questions/before_the_push_back"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :before_the_push_back, using: NewtonsSecondLaw do5  objects :body, :pusher6  axes :x7  instants :earlier_on, :applied, :later8  body.goes(:coasting, during: earlier_on..applied)9  body.goes(:pushed, during: applied..later)1011  take mass: body.mass12  take speed: body.velocity(at: applied)13  take force: body.force(by: pusher, during: pushed).magnitude14  take earlier: coasting.time_taken1516  deduce_from "directed towards the south", body.force(by: pusher, during: pushed).direction => -x17  deduce_from "x = 0", body.position(at: applied) => "0 m"18  deduce_from "constant speed", body.net_force(during: coasting) => "0 N"1920  position = body.position(at: earlier_on)21  answer position:, in: "m"2223  conclusion "Before the force acts the body just runs on at its steady speed, so the force has no say in " \24             "where it was."25end

playground.rb

1require_relative "../solutions/before_the_push_back"23play_with :before_the_push_back do4  title "Before the Push Back"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 body during coasting along x	$Δ x (c)$	m	—
position of body at applied along x	$x (applied)$	m	—
position of body at earlier on along x	$x (earlier_on)$	m	—
displacement of body during pushed along x	$Δ x (p)$	m	—
position of body at later along x	$x (later)$	m	—
average velocity of body during coasting along x	$v_{av} (c)$	m/s	—
time taken of coasting	$Δ t_{c}$	s	0 … 1e+05
average velocity of body during pushed along x	$v_{av} (p)$	m/s	—
time taken of pushed	$Δ t_{p}$	s	0 … 1e+05
average speed of body during coasting	$c_{av} (c)$	m/s	—
path length of body during coasting	$d (c)$	m	—
average speed of body during pushed	$c_{av} (p)$	m/s	—
path length of body during pushed	$d (p)$	m	—
change of velocity of body during coasting along x	$Δ v (c)$	m/s	—
velocity of body at applied from ground along x	$v (applied)$	m/s	—
velocity of body at earlier on from ground along x	$v (earlier_on)$	m/s	—
change of velocity of body during pushed along x	$Δ v (p)$	m/s	—
velocity of body at later from ground along x	$v (later)$	m/s	—
average acceleration of body during coasting along x	$a_{av} (c)$	m/s²	—
average acceleration of body during pushed along x	$a_{av} (p)$	m/s²	—
acceleration of body during coasting along x	$a (c)$	m/s²	—
acceleration of body during pushed along x	$a (p)$	m/s²	—
net force on body during coasting along x	$F_{net} (c)$	N	—
mass of body	$m$	kg	0 … ∞
net force on body during pushed along x	$F_{net} (p)$	N	—
mass of pusher	$m_{p}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on body	$\| W \|$	N	—
weight on pusher	$\| W_{p} \|$	N	—
force on body by pusher during pushed	$\| F_{p} (p) \|$	N	—
force on body by pusher during pushed along x	$F_{p} (p)$	N	—
displacement of body during coasting	$\| Δ x (c) \|$	m	—
position of body at applied	$\| x (applied) \|$	m	—
position of body at earlier on	$\| x (earlier_on) \|$	m	—
displacement of body during pushed	$\| Δ x (p) \|$	m	—
position of body at later	$\| x (later) \|$	m	—
average velocity of body during coasting	$\| v_{av} (c) \|$	m/s	—
average velocity of body during pushed	$\| v_{av} (p) \|$	m/s	—
change of velocity of body during coasting	$\| Δ v (c) \|$	m/s	—
velocity of body at applied from ground	$\| v (applied) \|$	m/s	—
velocity of body at earlier on from ground	$\| v (earlier_on) \|$	m/s	—
change of velocity of body during pushed	$\| Δ v (p) \|$	m/s	—
velocity of body at later from ground	$\| v (later) \|$	m/s	—
average acceleration of body during coasting	$\| a_{av} (c) \|$	m/s²	—
average acceleration of body during pushed	$\| a_{av} (p) \|$	m/s²	—
acceleration of body during coasting	$\| a (c) \|$	m/s²	—
acceleration of body during pushed	$\| a (p) \|$	m/s²	—
net force on body during coasting	$\| F_{net} (c) \|$	N	—
net force on body during pushed	$\| F_{net} (p) \|$	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 (c) = x (applied) - x (earlier_on)

change of position(coasting, x)

Δ x (p) = x (later) - x (applied)

change of position(pushed, x)

v_{av} (c) = \frac{Δ x (c)}{Δ t_{c}}

displacement a time(coasting, x)

v_{av} (p) = \frac{Δ x (p)}{Δ t_{p}}

displacement a time(pushed, x)

c_{av} (c) = \frac{d (c)}{Δ t_{c}}

path a time(coasting, x)

c_{av} (p) = \frac{d (p)}{Δ t_{p}}

path a time(pushed, x)

Δ v (c) = v (applied) - v (earlier_on)

velocity gained(coasting, x)

Δ v (p) = v (later) - v (applied)

velocity gained(pushed, x)

a_{av} (c) = \frac{Δ v (c)}{Δ t_{c}}

velocity gained a time(coasting, x)

a_{av} (p) = \frac{Δ v (p)}{Δ t_{p}}

velocity gained a time(pushed, x)

v (applied) = v (earlier_on) + a (c) \cdot Δ t_{c}

velocity after a time(coasting, x)

v (later) = v (applied) + a (p) \cdot Δ t_{p}

velocity after a time(pushed, x)

x (applied) = x (earlier_on) + v (earlier_on) \cdot Δ t_{c} + \frac{a (c) \cdot {Δ t_{c}}^{2}}{2}

position after a time(coasting, x)

x (later) = x (applied) + v (applied) \cdot Δ t_{p} + \frac{a (p) \cdot {Δ t_{p}}^{2}}{2}

position after a time(pushed, x)

{v (applied)}^{2} = {v (earlier_on)}^{2} + 2 \cdot a (c) \cdot (x (applied) - x (earlier_on))

velocity after a distance(coasting, x)

{v (later)}^{2} = {v (applied)}^{2} + 2 \cdot a (p) \cdot (x (later) - x (applied))

velocity after a distance(pushed, x)

x (applied) - x (earlier_on) = \frac{(v (applied) + v (earlier_on)) \cdot Δ t_{c}}{2}

mean of the velocities(coasting, x)

x (later) - x (applied) = \frac{(v (later) + v (applied)) \cdot Δ t_{p}}{2}

mean of the velocities(pushed, x)

F_{net} (c) = m \cdot a (c)

second law(coasting, x)

F_{net} (p) = m \cdot a (p)

second law(pushed, x)

F_{net} (p) = F_{p} (p) + 0

net force is the resultant(body, pushed, x)

| Δ x (c) | = \sqrt{{Δ x (c)}^{2}}

magnitude from components(displacement of body during coasting)

| x (applied) | = \sqrt{{x (applied)}^{2}}

magnitude from components(position of body at applied)

| x (earlier_on) | = \sqrt{{x (earlier_on)}^{2}}

magnitude from components(position of body at earlier on)

| Δ x (p) | = \sqrt{{Δ x (p)}^{2}}

magnitude from components(displacement of body during pushed)

| x (later) | = \sqrt{{x (later)}^{2}}

magnitude from components(position of body at later)

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

magnitude from components(average velocity of body during coasting)

| v_{av} (p) | = \sqrt{{v_{av} (p)}^{2}}

magnitude from components(average velocity of body during pushed)

| Δ v (c) | = \sqrt{{Δ v (c)}^{2}}

magnitude from components(change of velocity of body during coasting)

| v (applied) | = \sqrt{{v (applied)}^{2}}

magnitude from components(velocity of body at applied from ground)

| v (earlier_on) | = \sqrt{{v (earlier_on)}^{2}}

magnitude from components(velocity of body at earlier on from ground)

| Δ v (p) | = \sqrt{{Δ v (p)}^{2}}

magnitude from components(change of velocity of body during pushed)

| v (later) | = \sqrt{{v (later)}^{2}}

magnitude from components(velocity of body at later from ground)

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

magnitude from components(average acceleration of body during coasting)

| a_{av} (p) | = \sqrt{{a_{av} (p)}^{2}}

magnitude from components(average acceleration of body during pushed)

| a (c) | = \sqrt{{a (c)}^{2}}

magnitude from components(acceleration of body during coasting)

| a (p) | = \sqrt{{a (p)}^{2}}

magnitude from components(acceleration of body during pushed)

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

magnitude from components(net force on body during coasting)

| F_{net} (p) | = \sqrt{{F_{net} (p)}^{2}}

magnitude from components(net force on body during pushed)

| F_{p} (p) | = \sqrt{{F_{p} (p)}^{2}}

magnitude from components(force on body by pusher during pushed)

F_{p} (p) = - | F_{p} (p) |

directed towards the south

m \neq 0

has mass?(body)

m_{p} \neq 0

has mass?(pusher)

F_{net} (c) = 0

free of forces?(body, coasting)

F_{net} (p) = 0

free of forces?(body, pushed)