Stopped by a Steady Force

solved NewtonsSecondLaw

Q. A body of 20 kg $m = 20$ kg is moving at 15 m/s $v_{0} = 15$ m/s when a steady force of 50 N $| F_{net} | = 50$ N starts to act against its motion $F_{net} = - | F_{net} |$ . How long does it take to stop $v_{1} = 0$ m/s, in seconds?

Inputs

mass

m

20 kg

speed

v_{0}

15.0 m/s

force

| F_{net} |

50 N

Outputs

time

Δ t

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 20$ kg
$v_{0} = 15$ m/s
$| F_{net} | = 50$ N
$v_{1} = 0$ m/s from “to stop”
$F_{net} = - | F_{net} |$ from “against its motion”

Find $Δ t$ .

Working

against its motion $F_{net} = - | F_{net} |$
$F_{net} = - 50$
$F_{net} = −50$ N
second law(stopping, x) $F_{net} = m \cdot a$
$(−50) = 20 \cdot a$
$a = −2.5$ m/s²
velocity after a time(stopping, x) $v_{1} = v_{0} + a \cdot Δ t$
$0 = 15 + (−2.5) \cdot Δ t$
$Δ t = 6.00 s$

Conclusion

The force slows it at a steady rate, the force over the mass, so the time to stop is the speed it has to lose over that rate.

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 :stopped_by_a_steady_force do4  kind :numerical5  describes "A body of %{mass} is moving at %{speed} when a steady force of %{force} starts to act against " \6            "its motion. %{time}, in seconds?"7  given mass: "20 kg"8  given speed: "15 m/s"9  given force: "50 N"10  asks time: "How long does it take to stop"11end

solution.rb

1require_relative "../questions/stopped_by_a_steady_force"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :stopped_by_a_steady_force, using: NewtonsSecondLaw do5  objects :body6  axes :x7  body.goes(:stopping)89  take mass: body.mass10  take speed: body.initial_velocity11  take force: body.net_force.magnitude1213  deduce_from "against its motion", body.net_force.direction => -x14  deduce_from "to stop", body.final_velocity => "0 m/s"1516  time = stopping.time_taken17  answer time:, in: "s"1819  conclusion "The force slows it at a steady rate, the force over the mass, so the time to stop is " \20             "the speed it has to lose over that rate."21end

playground.rb

1require_relative "../solutions/stopped_by_a_steady_force"23play_with :stopped_by_a_steady_force do4  title "Stopped by a Steady Force"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 stopping along x	$Δ x$	m	—
position of body at end of stopping along x	$x_{1}$	m	—
position of body at start of stopping along x	$x_{0}$	m	—
average velocity of body during stopping along x	$v_{av}$	m/s	—
time taken of stopping	$Δ t$	s	0 … 1e+05
average speed of body during stopping	$c_{av}$	m/s	—
path length of body during stopping	$d$	m	—
change of velocity of body during stopping along x	$Δ v$	m/s	—
velocity of body at end of stopping from ground along x	$v_{1}$	m/s	—
velocity of body at start of stopping from ground along x	$v_{0}$	m/s	—
average acceleration of body during stopping along x	$a_{av}$	m/s²	—
acceleration of body during stopping along x	$a$	m/s²	—
net force on body during stopping along x	$F_{net}$	N	—
mass of body	$m$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on body	$\| W \|$	N	—
displacement of body during stopping	$\| Δ x \|$	m	—
position of body at end of stopping	$\| x_{1} \|$	m	—
position of body at start of stopping	$\| x_{0} \|$	m	—
average velocity of body during stopping	$\| v_{av} \|$	m/s	—
change of velocity of body during stopping	$\| Δ v \|$	m/s	—
velocity of body at end of stopping from ground	$\| v_{1} \|$	m/s	—
velocity of body at start of stopping from ground	$\| v_{0} \|$	m/s	—
average acceleration of body during stopping	$\| a_{av} \|$	m/s²	—
acceleration of body during stopping	$\| a \|$	m/s²	—
net force on body during stopping	$\| F_{net} \|$	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 = x_{1} - x_{0}

change of position(stopping, x)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(stopping, x)

c_{av} = \frac{d}{Δ t}

path a time(stopping, x)

Δ v = v_{1} - v_{0}

velocity gained(stopping, x)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(stopping, x)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(stopping, x)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(stopping, x)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(stopping, x)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(stopping, x)

F_{net} = m \cdot a

second law(stopping, x)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of body during stopping)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of body at end of stopping)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of body at start of stopping)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of body during stopping)

| Δ v | = \sqrt{{Δ v}^{2}}

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

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of body at end of stopping from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of body at start of stopping from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of body during stopping)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of body during stopping)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on body during stopping)

F_{net} = - | F_{net} |

against its motion

m \neq 0

has mass?(body)

F_{net} = 0

free of forces?(body, stopping)