Pushed Back, Then Left Alone

solved NewtonsSecondLaw

Q. A body of mass 0.40 kg $m = 0.40$ kg moving initially with a constant speed 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 $Δ t_{p} = 30$ 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 is its position at t = 100 s $Δ t_{s} = 100$ s, in kilometres?

Inputs

mass

m

0.4 kg

speed

v

10.0 m/s

force

| F_{p} |

8 N

duration

Δ t_{p}

30.00 s

time

Δ t_{s}

100.00 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
$Δ t_{p} = 30$ s
$Δ t_{s} = 100$ s
$x (applied) = 0$ m from “x = 0”
$F_{net} (l) = 0$ N from “for 30 s”
$F_{p} (p) = - | F_{p} (p) |$ from “directed towards the south”

Find $x (later)$ .

Working

directed towards the south $F_{p} (p) = - | F_{p} (p) |$
$F_{p} (p) = - 8$
$F_{p} (p) = −8$ N
net force is the resultant(body, pushed, x) $F_{net} (p) = F_{p} (p) + 0$
$F_{net} (p) = (−8) + 0$
$F_{net} (p) = −8$ N
second law(pushed, x) $F_{net} (p) = m \cdot a (p)$
$(−8) = 0.4 \cdot a (p)$
$a (p) = −20$ m/s²
position after a time(pushed, x) $x (released) = x (applied) + v (applied) \cdot Δ t_{p} + \frac{a (p) \cdot {Δ t_{p}}^{2}}{2}$
$x (released) = 0 + 10 \cdot 30 + \frac{(−20) \cdot 30^{2}}{2}$
$x (released) = −8700$ m
velocity after a time(pushed, x) $v (released) = v (applied) + a (p) \cdot Δ t_{p}$
$v (released) = 10 + (−20) \cdot 30$
$v (released) = −590$ m/s
time in parts(so far) $Δ t_{s} = Δ t_{p} + Δ t_{l}$
$100 = 30 + Δ t_{l}$
$Δ t_{l} = 70$ s
second law(left alone, x) $F_{net} (l) = m \cdot a (l)$
$0 = 0.4 \cdot a (l)$
$a (l) = 0$ m/s²
position after a time(left alone, x) $x (later) = x (released) + v (released) \cdot Δ t_{l} + \frac{a (l) \cdot {Δ t_{l}}^{2}}{2}$
$x (later) = (−8700) + (−590) \cdot 70 + \frac{0 \cdot 70^{2}}{2}$
$x (later) = −50$ km
$x (later) = −50000$ m $= −50$ km
$x (later) \approx −50$ kmto 2 significant figures

Conclusion

Once the force stops, the body keeps the velocity it was left with, so after the push it runs on south at a steady speed.

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 :pushed_back_then_left_alone 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 %{duration}. 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} at t = %{time}, " \8            "in kilometres?"9  given mass: "0.40 kg"10  given speed: "10 m/s"11  given force: "8.0 N"12  given duration: "30 s"13  given time: "100 s"14  asks position: "What is its position"15end

solution.rb

1require_relative "../questions/pushed_back_then_left_alone"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :pushed_back_then_left_alone, using: NewtonsSecondLaw do5  objects :body, :pusher6  axes :x7  instants :applied, :released, :later8  body.goes(:pushed, during: applied..released)9  body.goes(:left_alone, during: released..later)10  body.goes(:so_far, during: applied..later)1112  take mass: body.mass13  take speed: body.velocity(at: applied)14  take force: body.force(by: pusher, during: pushed).magnitude15  take duration: pushed.time_taken16  take time: so_far.time_taken1718  deduce_from "directed towards the south", body.force(by: pusher, during: pushed).direction => -x19  deduce_from "x = 0", body.position(at: applied) => "0 m"20  deduce_from "for 30 s", body.net_force(during: left_alone) => "0 N"2122  position = body.position(at: later)23  answer position:, in: "km", figures: 22425  conclusion "Once the force stops, the body keeps the velocity it was left with, so after the push it runs " \26             "on south at a steady speed."27end

playground.rb

1require_relative "../solutions/pushed_back_then_left_alone"23play_with :pushed_back_then_left_alone do4  title "Pushed Back, Then Left Alone"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 pushed along x	$Δ x (p)$	m	—
position of body at released along x	$x (released)$	m	—
position of body at applied along x	$x (applied)$	m	—
displacement of body during left alone along x	$Δ x (l)$	m	—
position of body at later along x	$x (later)$	m	—
displacement of body during so far along x	$Δ x (s)$	m	—
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 velocity of body during left alone along x	$v_{av} (l)$	m/s	—
time taken of left alone	$Δ t_{l}$	s	0 … 1e+05
average velocity of body during so far along x	$v_{av} (s)$	m/s	—
time taken of so far	$Δ t_{s}$	s	0 … 1e+05
average speed of body during pushed	$c_{av} (p)$	m/s	—
path length of body during pushed	$d (p)$	m	—
average speed of body during left alone	$c_{av} (l)$	m/s	—
path length of body during left alone	$d (l)$	m	—
average speed of body during so far	$c_{av} (s)$	m/s	—
path length of body during so far	$d (s)$	m	—
change of velocity of body during pushed along x	$Δ v (p)$	m/s	—
velocity of body at released from ground along x	$v (released)$	m/s	—
velocity of body at applied from ground along x	$v (applied)$	m/s	—
change of velocity of body during left alone along x	$Δ v (l)$	m/s	—
velocity of body at later from ground along x	$v (later)$	m/s	—
change of velocity of body during so far along x	$Δ v (s)$	m/s	—
average acceleration of body during pushed along x	$a_{av} (p)$	m/s²	—
average acceleration of body during left alone along x	$a_{av} (l)$	m/s²	—
average acceleration of body during so far along x	$a_{av} (s)$	m/s²	—
acceleration of body during pushed along x	$a (p)$	m/s²	—
acceleration of body during left alone along x	$a (l)$	m/s²	—
acceleration of body during so far along x	$a (s)$	m/s²	—
net force on body during pushed along x	$F_{net} (p)$	N	—
mass of body	$m$	kg	0 … ∞
net force on body during left alone along x	$F_{net} (l)$	N	—
net force on body during so far along x	$F_{net} (s)$	N	—
acceleration of body during pushed	$\| a (p) \|$	m/s²	—
acceleration of body during so far	$\| a (s) \|$	m/s²	—
acceleration of body during left alone	$\| a (l) \|$	m/s²	—
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 pushed	$\| Δ x (p) \|$	m	—
position of body at released	$\| x (released) \|$	m	—
position of body at applied	$\| x (applied) \|$	m	—
displacement of body during left alone	$\| Δ x (l) \|$	m	—
position of body at later	$\| x (later) \|$	m	—
displacement of body during so far	$\| Δ x (s) \|$	m	—
average velocity of body during pushed	$\| v_{av} (p) \|$	m/s	—
average velocity of body during left alone	$\| v_{av} (l) \|$	m/s	—
average velocity of body during so far	$\| v_{av} (s) \|$	m/s	—
change of velocity of body during pushed	$\| Δ v (p) \|$	m/s	—
velocity of body at released from ground	$\| v (released) \|$	m/s	—
velocity of body at applied from ground	$\| v (applied) \|$	m/s	—
change of velocity of body during left alone	$\| Δ v (l) \|$	m/s	—
velocity of body at later from ground	$\| v (later) \|$	m/s	—
change of velocity of body during so far	$\| Δ v (s) \|$	m/s	—
average acceleration of body during pushed	$\| a_{av} (p) \|$	m/s²	—
average acceleration of body during left alone	$\| a_{av} (l) \|$	m/s²	—
average acceleration of body during so far	$\| a_{av} (s) \|$	m/s²	—
net force on body during pushed	$\| F_{net} (p) \|$	N	—
net force on body during left alone	$\| F_{net} (l) \|$	N	—
net force on body during so far	$\| F_{net} (s) \|$	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 (p) = x (released) - x (applied)

change of position(pushed, x)

Δ x (l) = x (later) - x (released)

change of position(left alone, x)

Δ x (s) = x (later) - x (applied)

change of position(so far, x)

v_{av} (p) = \frac{Δ x (p)}{Δ t_{p}}

displacement a time(pushed, x)

v_{av} (l) = \frac{Δ x (l)}{Δ t_{l}}

displacement a time(left alone, x)

v_{av} (s) = \frac{Δ x (s)}{Δ t_{s}}

displacement a time(so far, x)

c_{av} (p) = \frac{d (p)}{Δ t_{p}}

path a time(pushed, x)

c_{av} (l) = \frac{d (l)}{Δ t_{l}}

path a time(left alone, x)

c_{av} (s) = \frac{d (s)}{Δ t_{s}}

path a time(so far, x)

Δ v (p) = v (released) - v (applied)

velocity gained(pushed, x)

Δ v (l) = v (later) - v (released)

velocity gained(left alone, x)

Δ v (s) = v (later) - v (applied)

velocity gained(so far, x)

a_{av} (p) = \frac{Δ v (p)}{Δ t_{p}}

velocity gained a time(pushed, x)

a_{av} (l) = \frac{Δ v (l)}{Δ t_{l}}

velocity gained a time(left alone, x)

a_{av} (s) = \frac{Δ v (s)}{Δ t_{s}}

velocity gained a time(so far, x)

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

velocity after a time(pushed, x)

v (later) = v (released) + a (l) \cdot Δ t_{l}

velocity after a time(left alone, x)

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

velocity after a time(so far, x)

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

position after a time(pushed, x)

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

position after a time(left alone, x)

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

position after a time(so far, x)

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

velocity after a distance(pushed, x)

{v (later)}^{2} = {v (released)}^{2} + 2 \cdot a (l) \cdot (x (later) - x (released))

velocity after a distance(left alone, x)

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

velocity after a distance(so far, x)

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

mean of the velocities(pushed, x)

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

mean of the velocities(left alone, x)

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

mean of the velocities(so far, x)

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

second law(pushed, x)

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

second law(left alone, x)

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

second law(so far, x)

d (s) = d (p) + d (l)

path in parts(so far)

Δ t_{s} = Δ t_{p} + Δ t_{l}

time in parts(so far)

| a (p) | = | a (s) |

steady through its parts(so far, 1)

| a (l) | = | a (s) |

steady through its parts(so far, 2)

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

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

| Δ x (p) | = \sqrt{{Δ x (p)}^{2}}

magnitude from components(displacement of body during pushed)

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

magnitude from components(position of body at released)

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

magnitude from components(position of body at applied)

| Δ x (l) | = \sqrt{{Δ x (l)}^{2}}

magnitude from components(displacement of body during left alone)

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

magnitude from components(position of body at later)

| Δ x (s) | = \sqrt{{Δ x (s)}^{2}}

magnitude from components(displacement of body during so far)

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

magnitude from components(average velocity of body during pushed)

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

magnitude from components(average velocity of body during left alone)

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

magnitude from components(average velocity of body during so far)

| Δ v (p) | = \sqrt{{Δ v (p)}^{2}}

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

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

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

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

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

| Δ v (l) | = \sqrt{{Δ v (l)}^{2}}

magnitude from components(change of velocity of body during left alone)

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

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

| Δ v (s) | = \sqrt{{Δ v (s)}^{2}}

magnitude from components(change of velocity of body during so far)

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

magnitude from components(average acceleration of body during pushed)

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

magnitude from components(average acceleration of body during left alone)

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

magnitude from components(average acceleration of body during so far)

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

magnitude from components(acceleration of body during pushed)

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

magnitude from components(acceleration of body during left alone)

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

magnitude from components(acceleration of body during so far)

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

magnitude from components(net force on body during pushed)

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

magnitude from components(net force on body during left alone)

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

magnitude from components(net force on body during so far)

| 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} (p) = 0

free of forces?(body, pushed)

F_{net} (l) = 0

free of forces?(body, left alone)

F_{net} (s) = 0

free of forces?(body, so far)